Just to fill in the expected argument...
The issue with mindsight is that, while it requires telepathy, and the Normal section implies that it uses telepathy, the actual benefit of the feat never says that is a form of telepathy, or that it is mind-affecting. It’s also not defined as a divination. Therefore, by the rules strictly as written, mindsight works, even if you have immunity to telepathy, to mind-affecting abilities, or to divination (e.g. you are undead or have mind blank).
This is because a feat’s requirement need not actually be related to how a feat functions (e.g. you don’t need to use Combat Expertise to benefit from Improved Trip), and the Normal section is explicitly treated as “reminder text,” to use a term from that other Wizards of the Coast product, and has no rules weight (e.g. even though the Normal section on Tower Shield Proficiency claims that nonproficiency results in taking the shield’s ACP as a penalty to all skills relating to movement, that’s not what actually happens because the “real rule,” given in the Armor section, is that all Strength- and Dexterity-based skill and ability checks take the penalty). Thus, there simply is no rule that mindsight is or relies on telepathy.
In practice, in all but the strictest games, this oversight it ignored, and lots of things protect against mindsight, such as immunity to telepathy, mind-affecting abilities, or divinations. Many also extend the Darkstalker feat to cover mindsight (and the lifesight feat from Libris Mortis). As Sandwich pointed out in a previous version of his answer, the Madness ability of allips would also form a serious deterrent to the mindseer when Mindsight is treated as telepathy, as the mindseer would quickly lose all Wisdom.
Strict-RAW defenses
Even under the rules read as strictly as possible, there are still some defenses against Mindsight.
Mindlessness
Mindless creatures are immune to Mindsight. Not terribly helpful for players since you can’t play a mindless creature, but it’s worth keeping in mind, particularly as a DM. Note that intelligent creatures that are usually immune to mind-affecting things, like intelligent undead, are not immune under this super-strict RAW.
Hellbreaker
If the mindseer is put inside a hellbreaker’s Telepathic Static, as Matthieu M. discovered, it loses telepathy, and can no longer use the Mindsight feat as it has lost the requirement.
Tower Shields
As Forrestfire just pointed out to me, when you have Total Cover (as when employing a tower shield), creatures cannot draw line of effect to you. Normally, this doesn’t matter much since touch attacks and targeted spells still work against the tower-shield user, but Mindsight is not targeted, so the shield somehow blocks the Mindsight, despite being far thinner than the requisite blocking material. This is really just an abuse of the terrible wording on that use of the tower shield, more than anything else.
Cerebral Blind
As Sandwich found, the Cerebral Blind class feature of the slayer class can be read as blocking all effects that reveal location. Note that the wording is somewhat ambiguous.
The other player is incorrect.
If you have telepathy, you can communicate with any creature that knows any language, regardless of whether or not you speak that language. As you quoted,
A creature with [telepathy] can communicate telepathically with any other creature [...] that has a language.
That is the only requirement: a language. Not any particular one, not one shared between you, just capable of language in general. It’s unclear whether it works on creatures that understand but cannot speak any languages, but that’s a corner-case not relevant to the usual uses of telepathy.
Best Answer
I treat telepathy just like speaking; it doesn't require an action to establish a link or whatnot. I could see requiring one if you want to start telepathically 'talking' to someone you can't see to represent the "mentally searching them out" part, and similarly if they go beyond 100' and then come back the telepath doesn't have any way except for normal sight or keeping trying to reach out telepathically to know that.
In general I agree with @PeterSeckler, interpret it whatever cool way you want and if PCs are finding some way to abuse it, toss in some limiters. I know a group would love a "can't be overheard and can work through walls" tactical coordination tool, but a little bit of "the rogue got dragged out of range/fell through a long pit trap" and "a psionic monster that can overhear telepathy just heard all your plans" from time to time compensates.