I'm not adopted, but I would like to say what I think.
I think the word "adoptee" is fine when you're talking about adoption. I don't think anyone needs to be introduced as "Here is Jan, the adoptee" or anything like that in other circumstances. However, when discussing adoption, there are the relinquishers, adopters, and adoptees... it's just grammar.
It would be wonderful if no one ever felt the need to say the "adopted son" or "adopted daughter" or things like that. However, I am very guilty of it myself. Growing up I had two adopted brothers, and whenever I introduced them, I made very sure that people knew they were my ADOPTED brothers. I think, however, that this was due to their particular situation. Both my brothers were mentally retarded, but not from downs syndrome or anything like that, they were mentally retarded because they had severe Fetal Alchohol Syndrome. Anytime anyone met them, they definitely acted very differently. Espescially when you're a kid, you have to say "Oh, that's because they have MR".... then people ask what that is, and you say "they're retarded" and then kids as "why"... and so on. It really wouldn't be fair to let people think that our mother (my biological, they're adoptive) had been so irresponsible as to harm her children by drinking to oblivion with them in the womb. So.. they are my "adopted" brothers - and obviously often even my parent's "adoped" sons. It's sad... but that's the way it is.
On the other hand, my best friend had a brother who was a adopted, and I never even knew until I got old enough to figure out there wasn't enough time between their birthdays. He was a perfectly normal kid, so his adoption was never even brought up. It wasn't a secret, it just wasn't relevant.
Because of this, as I've gotten older I have made a more calculated effort not to call my brothers "adopted" unless I get into having to explain their medical situation.
On a side note - both my "adopted" brothers (here I have to specify again because I have three others that were biological) have the same biological parents. They are actually the third and fifth child born to that couple. Because of the specific situations (abuse/foster care), they never lived together before we adopted them. The older one was 3 and a half when he was adopted, an the younger one was not quite two. They have both known they were adopted from day 1, and we often would visit they one other biological brother we could track down. The older of the pair has never shown the slightest unhappiness about being adopted, or even curiousity about who his biological parents were. The younger of the pair has obsessed about it pretty much since he started talking. I guess that just goes to show that no two people (even those with both sets of parents shared) feel the same way about being adopted.