Speaking, No. Understanding, Maybe.
The few Familiars that can automatically speak, do so "as a supernatural ability."
From PFSRD:
Familiar Basics
Skills:
[...] Regardless of a familiar's total skill modifiers, some skills may remain beyond the familiar's ability to use.
and further down that page:
Intelligent Animals
Increasing an animal's Intelligence to 3 or higher means it is smart enough to understand a language. However, unless an awaken spell is used, the animal doesn't automatically and instantly learn a language, any more than a human child does. The animal must be taught a language, usually over the course of months, giving it the understanding of the meaning of words and sentences beyond its trained responses to commands like "attack" and "heel."
Even if the animal is taught to understand a language, it probably lacks the anatomy to actually speak (unless awaken is used). For example, dogs, elephants, and even gorillas lack the proper physiology to speak humanoid languages, though they can use their limited "vocabulary" of sounds to articulate concepts, especially if working with a person who learns what the sounds mean.
So, to expand my answer a little... No they can't speak it, unless the animal has the right anatomy to produce speech. They can be taught to understand languages, though.
(You could argue that the gaining of a point of Linguistics reflects the months of learning a language, so that being Awakened is not a requirement.)
Familiars automatically meet the Intelligence requirement:
Familiar Ability Descriptions
All familiars have special abilities (or impart abilities to their masters) depending on the master's combined level in classes that grant familiars, as shown on the table below. The abilities are cumulative.
Master Class Level Natural Armor Adj. Intelligence
1st–2nd +1 6
...
The rules as written do seem unclear. It's also narratively ambiguous. While it would seem weird for becoming a familiar to make a creature stupider, it could also be explained as "only particularly stupid imps/pseudodragons/etc are inclined to become familiars to magic users of such low level that their intelligence score ought to be lower than base for their species."
On the other hand, it makes equally good sense to read the intelligence-at-level column for familiars as an endowment only and never a penalty. If it were intended to ever function as a penalty rather than a bonus, it seems there'd be a line about the possibility in the text of the rules and there simply isn't.
It's important to note that the improved familiar feat is an afterthought to the familiar mechanics more generally. In the base case of familiars, this question never comes up, it is only an issue when the improved familiar feat is applied. It would be unusual, at least, to override the improved familiar's base stats to its detriment according to the rules defining base familiars.
RAW is unclear, but I think available evidence points to authorial intent that familiarhood should only ever increase a creature's intelligence.
Best Answer
The designer did not intend for familiar archetypes to be applied to improved familiars
While I agree that it kind of sucks, Familiar Folio author Mark Seifter confirms as much in this Paizo messageboard thread. He says, in response to another poster, that
There are, nonetheless, probably ways to get the archetypes on an improved familiar anyway; Pathfinder is pretty vast, after all. But such ways aren't intentional.