Hi Valerie,
Funny you should bring this up. Within the past couple of years, I ran into a situation similar to the hypothetical you've described.
I am a competitive pool player in a national league. While I am much more flamboyantly girly than most of the female league players, I'm pretty well stealth in league and in my life generally. And I do like it that way.
A couple of years ago, I met another player, I'll call her Ann. Ann is a great pool player, better than I am. She's been playing league pool for nearly as long as I've been alive. Many league players knew Ann long before she transitioned; it is generally known among regional players that she is a trans woman. And to be honest, no one would look at her and assume that she's a cis woman. She seems to have accepted this.
Well, when I first met Ann, she was looking to get on a team. Even though she's a very good player, no one would pick her up. Some people in league openly refuse to play with her because she's transsexual. As a team captain, I invited Ann to play on my team.
Ann and I had been playing together for a couple of months before I decided to out myself to her. (She had no idea that I'm trans.) I think partly I wanted to feel some camaraderie with a fellow gender outlaw, and partly I wanted to demonstrate to Ann that she isn't alone.
Fast forward to today. To make a long story short, Ann turned out to be a not so nice person. I found her very difficult to manage as a team member. She was never where she needed to be when it was time for her to play, so I found myself wandering around often looking for her. She just had a lack of courtesy and a sense of being better than the rest of us. But that isn't the worst of it.
For reasons that I can't explain, Ann took to saying very nasty things to me. I don't know why; I was only ever nice to her. As the only 2 trans women that I know of in our regional league, I wanted to be her friend. I've come to understand that Ann is a very angry, self hating woman. For some of the people who don't want to play with her, the issue really is discrimination. But I think that for the majority of people who don't want to play with her, it's because she's an ass.
If Ann has outed me to anyone else, I'm unaware of it. I do, however, regret outing myself to her. I don't like her. I don't want to play with her. And given her history of randomly saying mean, hurtful things to me out of the blue and completely unprovoked, I don't trust that she will honor my confidence among fellow league players into the future.
I'm not saying that you shouldn't trust other trans people. What I am suggesting is that we as trans women are as varied in personality and honor as is the general non-trans population. Whether I will out myself to any future transsexuals that may show up at pool league is an open question. I will, however, wait to become friends first before revealing intimate details about my life.
As to the second part of your question, playing on a team with Ann did not cause anyone else in league to question my gender identity. I am very confident in my true gender, and people have no cause to question it. True, my hands are a little big for a woman, and I do have a bit of an Adam's apple, but everything else about me screams girly-girl. And I'm pretty sure that my own confidence causes people to simply accept me at face value.
Sorry, I didn't mean to write a book here. I hope I addressed the gist of your question.