Identifying a scroll is different from deciphering a scroll and neither requires expending the scroll.
Identifying
The detect magic spell determines if there're any magic auras in the area on the first round, the number of magic auras and the most potent aura in the area on the second, and where exactly those auras are and how powerful each is on the third. With a successful Knowledge (arcana) skill check (DC 15 + spell level), the caster can determine the aura's school of magic; this check apparently takes no time and is part of the 3rd-round detect magic effect. Doing this would tell the caster only the school of the spell or spells on the scroll.
A 3-round thorough examination of the object while employing the detect magic spell permits the caster to "[i]dentify the properties of a magic item" with a successful Spellcraft check (DC 15 + scroll's caster level). This tells the caster everything about the magic item, with apparently one exception. The caster knows whether the scroll is arcane or divine, what level spell is on it, what level the spell or spells are scribed at, and so on, but identifying doesn't reveal exactly what spell is on the scroll; to do that the caster must decipher the scroll.
These 3 rounds are in addition to the first 3 rounds leading to pinpointing each aura in the area of the spell detect magic. Events happen in order unless stated to happen simultaneously.
(The analyze dweomer spell also reveals these properties, but the word dweomer is difficult to say without laughing.)
Deciphering
To find out which spell is on the scroll (and subsequently be able to activate it) the caster must decipher the scroll, which requires a successful full-round Spellcraft skill check (DC 20 + spell level), a successful 1-minute Use Magic Device skill check (DC 25 + spell level), or the read magic spell.
After the scroll's deciphered, that scroll's always deciphered for that caster. No further checks need be made, and the scroll can be employed normally.
Don't conflate identifying with deciphering. The hairs are fine and difficult to split but the game splits them anyway: When a caster identifies an item's properties, he gets command words, number of charges, and so on (pretty much just like he read the item's description from the book) but not the name of the spell on the scroll; the caster gets that and the ability to activate the scroll from deciphering the scroll.
Note: That's weird and the Pathfinder Role-playing Game buries that on page 490: "The writing on a scroll must be deciphered before a character can... know exactly what spell it contains." This also isn't exclusive to Pathfinder--both the Dungeons and Dragons 3.0 and 3.5 Dungeon Master's Guides have the exact same language in their Deciphering a Scroll sections (pages 203 and 238, respectively).
Note: A solid but dangerous case can be made for skipping the identification step if the caster already knows what he's dealing with, much in the same way someone proficient with a sword can just whack fools with the sword instead of learning its maker, properties, and history before he starts using it. The DM rolls secretly to determine if a character deciphers gibberish via the Linguistics skill, but the player rolls openly to decipher a magic scroll via the Spellcraft skill or Use Magic Device skill, so right there the player will know it's a magic scroll and, if successful in deciphering it, know the spell on it. The character can then try to activate the scroll, but--and this is really a DM's call--the character still won't know any of the scroll's properties (such as it's spell level or the scriber's caster level) because the character's not identified the scroll's properties. Just as Fighter Joe can still take swings with a magic sword without knowing its exact properties, Wizard Jim can decipher a scroll and, if he meets the requirements, cast the spell from the scroll without knowing exactly what's going to happen. I've never seen a player want to do that, but I'd totally let him. That could be hilarious.
Other Tidbits: There are spells that straight-up answer questions and get information; one could possibly use those and make a big pile of information about a scroll. There are feats and classes that are scroll-specific, and those might be useful. I've ignored both because they seem to be beyond the question's scope.
1. No, they don't have a spellbook
Jeremy Crawford (official rules guru for D&D) has said this very clearly:
An Arcane Trickster casts wizard spells but doesn't use a spellbook.
The same is true of the Eldritch Knight.
The only class with a spellbook is the wizard. There is nothing in the Arcane Trickster's description that mentions anything about a spellbook so there is no reason to assume it would get one.
Arcane Trickers may learn wizard spells but they are not wizards.
For example the Arcane Trickster feature says that they know spells:
You know three 1st-level wizard spells of your choice...
In essence, the Arcane Trickster knows the few spells so well that they have them always in mind. Knowing spells mean there is no need for a spellbook and no way to use one even if they did have one.
This is directly opposed to way the wizard must prepare their spells from their spellbook.
2. No, they don't have a spellbook
(see 1.)
3. No, they cannot learn spells like a wizard
See this excellent answer for a full discussion.
The Wizard is the only class to have a method to learn new spells aside from level-up or feats. And again the Arcane Trickster is not a wizard in any way.
The Arcane Trickster only lists one way to learn new spells: leveling up.
The Spells Known column of the Arcane Trickster Spellcasting table shows when you learn more wizard spells...For instance, when you reach 7th level in this class, you can learn one new spell of 1st or 2nd level.
Note: Arcane Tricksters can use spell scrolls whose spells are on the wizard list
Jeremy Crawford confirmed this:
The intent is yes, the Arcane Trickster and the Eldritch Knight can use scrolls with wizard spells on them.
But they definitely cannot learn spells from them.
Best Answer
Disclaimer
Some of the possible solutions I'm about to propose only work for certain methods the Imposter could be using for the deception; for example, they might have replaced the scroll your players are using with one under their control, or they might have stolen the original other scroll. There are too many possibilities to discuss them all, but I'll try to briefly mention what kinds of deceptions each potential solution will be effective against.
Visions of the Past
If the scroll you have was ever in the Imposter's possession, then the Knowledge Cleric's Visions of the Past feature should reliably help identify them. The relevant portions of the feature are:
Your party doesn't have a Cleric, let alone a 17th level Knowledge domain one, but perhaps they could go on a quest to find one and earn their services.
Locate Object
If the Imposter is nearby, such as in the same city, it may be possible to use Locate Object to find the other half of the pair of scrolls. This is a bit situational and requires a bit of DM ruling, but I would probably allow it. Here is the relevant text:
The DM would have to rule how specific a "particular kind" of object can be, but if you put your own scroll into a lead-lined box then locating another of that "kind" of object should be possible. Since it's a 2nd level spell with no costly components and it's available to your Paladin and Druid, and possibly your Wizard, you could cast it multiple times in a day to cover a larger city. It would be very impractical to search much further than that though.
Scrying
Scrying is very dependent on the DM ruling that "whoever is writing these messages" is sufficient to target someone, but if it's allowed then Scrying should work on a wide range of possible Imposter methods. There are some complications though. Scrying requires a focus costing 1000gp, which might be quite expensive for your party. It also allows for a saving throw, and since we're already stretching whether the target is even valid they would definitely get the bonus to their save from being not very familiar. Additionally, a successful save means you can't try again for 24 hours. Finally, your party won't gain access to the spell until level 9. All that together means it could take quite a while to actually get a result, though nothing about the spell's description indicates that the target will know about attempts whether successful or not so that might not be a problem.
Skill Checks
When players want to do something that doesn't necessary have official rules, it's appropriate to call for a skill or tool check (or technically an ability check that lets you add your proficiency bonus if you have proficiency in a particular skill or tool). Here are a few examples of checks that could be appropriate based on examples from the PHB section on ability checks and the Xanathar's Guide section on tool proficiencies with some emphasis added:
An Intelligence (Arcana) or Intelligence (Forger's Kit) check to determine that someone has messed with the magic of the item directly:
An Intelligence (Investigation) or Intelligence (Forger's Kit) check to determine an Impostor has forged the scroll itself or is imitating your contact's handwriting:
A Wisdom (Insight) check to determine whether the contact has changed writing style subtly that could indicate they were replaced:
Your players could also seek help from a Forger NPC if they aren't proficient in relevant tools and skills.
Divine Help
This option is the most dependent on DM rulings, and as a consequence has a lot of potential flexibility and could even be used for more indirect Imposter situations such as mind control or shapeshifters that even Scrying might miss because the writer doesn't look different. Some examples of divine help are the spells Divination and Commune, and the 10th level Cleric feature Divine Intervention. All of these have the potential to gain relevant information from a deity with the right wording, but it's up to the DM exactly what information the deity has and how they choose to share it.
None of your players naturally has access to these even at higher levels, so if you want these to be an option then you'll need to decide how they're accessible as well.