If the Collectors don't remain avatars of their respective Reaper like I suspect the Husks will do, they'll be set free by the Synthesis, and they'll eventually become aware of their origin and history. No idea what happens then, but I can't imagine them becoming a regular species of the galaxy.
This is one reason I prefer Control, as I can repair the Collectors and hopefully improve their ability to think freely. I do believe they have intelligence and limited culture of a sort: they have a language; their technology possesses very distinct visual styles, even distinct from the Reapers, themselves, implying preferences; Harbinger, when possessing one Collector on Horizon, makes a speech with gestures, which would be wholly pointless if the Collectors were just telepathically controlled slaves. I suspect they don't have much free will or individuality, but I don't know how different that is from, say, the rachni. The Collectors certainly aren't Protheans anymore, but they're not without potential... though I'd like to give them a digestive tract back.
As a sidenote, I'm a bit annoyed with the Prothean implementation in ME3. In ME2, one of the Cerberus crew says something like "I thought the Protheans would be more regal. Not giant... bugs." This is an aspect I actually liked, because it was probably quite a departure from how many would likely think of the Protheans... until ME3 showed them to be more conventional reptile/mammal aliens and that the Reapers made them insectoid for... reasons unknown. I'm not sure it was a great idea to make a species alien enough that it didn't even exist in the same time period be less alien.
I don't think they regain their intelligence. The Collectors are fully indoctrinated slaves...and they were indoctrinated for a very long time. We know Indoctrination seems to destroy or at least severely deteriorate people's cognitive functions once it advances far enough...that's not curable.
It should be curable with precise enough brain surgery. We might not have the technology for it yet, but that shouldn't make it impossible. I rather suspect Synthesis' upgrades could cure indoctrination, for instance.