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Fans have, of course, always liked to know what each other were thinking.
See many Western media fandom interview articles here on Fanlore.
A few print zines included interviews, but the practice of formal interviews appears to be an online activity and something that most popular between 1996 and the mid to late 2000s. Some of these interviews are online as text and image, as audio-only, and as video.
Also see Oral History.
Why Fan Interviews?
These fan interviews served a number of purposes -- social interaction, self-promotion, discussion of meta are only a few.
Most online interviews were done through a fannish central site, and usually as a component of a fiction archive and other fan resources. This means there is a fair amount of repetition in the questions asked, and in the answers received; the interviewee had access to others' answers before responding. This also means there was a bit of self-policing regarding the answers, as it is human nature to not want to rock the boat, at least in one's current fannish home.  Many fans' replies are not very controversial, or critical of fandom, unless they were trying for shock value, or if the interviewee had a reputation of being a wild card, something which needed to be upheld. There are a few X-Files Fan Interviews that fit that bill.
In the present day, the practice of fans formally interviewing each other appears to be happening at a slower pace. This may be due to the rise of Live Journal, Dreamwidth, and blogs. Everyone's opinion can now hang out all over the place and one is no longer dependent on the gatekeepers of central fannish websites for a voice and venue. An exception to this trend are podcasts, a platform that picks up the trend a bit.
In any case, these interviews are fascinating and fun for a number of reasons. They touch on the topics of hopes and fears, discuss then-current trends, and are full of enthusiastic discussion, often shedding light on things that fans today may have forgotten. The interviews are also filled with the language and slang of the day, sometimes providing insight into current fandom terms. It is also very interesting to read of fans' reactions to coming technological trends and realities. And, they are often examples of how well-spoken, witty, self-aware, and smart fans can be.
Recurring Tropes: "Good Riddance" and "It's Better Here Than in That Last Place"
A statement many fans make in interviews is that their first works were "dreadful," "amateurish," "embarrassing," and likely "Mary Sues." Fans often made a point of saying that they were glad their juvenilia had been lost somewhere on the internet or erased in some way. One example: "Somewhere in the landfill, there's probably some really terrible poetry I wrote about Star Trek when I was fifteen, which—thank God—is in a landfill." 
Another trend: fans also tend to downplay the value and quality of fandoms they were in previously, stating that the fandom they were in now was superior: better fiction, better art, and better company.
Fandoms Without Many Interviews
Some bigger fandoms never developed a robust "interview culture." Some examples:
- Blake's 7 was old enough and had a major central venue at Judith Proctor's Blake's 7 site, but the timing of the internet, and possibly the Blake's 7 Wars, curbed this activity.
- Starsky & Hutch is certainly an old enough fandom and straddled some of the technology, but perhaps too insular by the time the internet picked up steam. The Starsky & Hutch Fan Interviews tend to be quite bland and short, due in part to the fact that the fandom saw itself as the "fandom of nice," perhaps because many of them were hosted on a gen-only fiction site.
- The Professionals is an example of a long-running and opinionated fandom, but its small number of interviews were in letterzines and meant to be read by only a few. Some of those interviews were posted to The Hatstand in later years.
- Some fandoms, while large and covering a large amount of time were, were/are simply too spread out to have a central host, Star Trek Fan Interviews is one example; there are many, but not as many interviews as one would expect.
- The Forever Knight fandom is one that appears to have no fan-to-fan interviews, something that may be explained by the fact that this activity just never gained any traction there.
Fans also have had a long history of interviewing actors, showrunners, and other TPTB. For examples of fans being interviewed by non-fans, see: Category:News Media, Category:Academic Commentaries, and Category:Perspectives on Fans.