It is possible to do that. In fact... If you ever wanted to train as a psychoanalysis then a requirement of that is that the prospective clinician successfully complete a course of analysis. The majority of other therapies don't have that requirement, but some therapies encourage clients to have their own therapists - to discuss counter-transference responses and stuff like that.

That being said IMHO there can be more stigma from within the professions that deal in mental health issues than there is without. I'd keep it under your hat as much as possible (though that is just my personal opinion and I'm aware that others have been very upfront about it and that has worked out alright for them).

There are a variety of opinions out there. Some people think that it is inappropriate for people with certain conditions to be therapists (schizophrenia, bi-polar, and personality disorders most notably). Other people think that it is fine so long as you are acting in a professional matter. Other people think that if you have ever received a psychiatric dx then you shouldn't do that. Some other people think that past dx is okay so long as you don't meet current criteria...

Lots of people go into therapy because of their experiences in therapy. There are some psychiatrists (successful ones) who have similarly gone into psychiatry because of their experiences in the mental health system. That being said there is still a lot of prejudice out there. My motto is: It is none of their business. The only thing that is their business is whether I am able to conduct myself and treat my clients in a competent professional fashion.