As far as work in concerned, I don't talk about where I work on the internet though any of my public personas or on stage (I do storytelling from time to time). Likewise, any conversations about my personal life at the office is strictly on a "need to know" basis. That being said, I have just finished a two-year term with our diversity council and plan to be more active supporting the LGBT affinity group this year. While I am not gay (relative to my birth gender that is), I do think there may be some value in sharing that I'm TG with that group. A lot of members of the group are either straight "allies" or "under the radar". That's not to say they are closeted, they're just concerned about potential impact to their careers. Sadly, LGBT still carries a negative stigma with those in position to make promotion/hiring decisions.
Yet, if no one was out at work, there would be no LGBT support group. Additionally, the "T" part of "LGBT" tends to be under-represented. I have outed myself to a long-time friend who works at the same place, partially to explain my empathy toward the LGBT group and partially because it was weird that she'd known me so long and never knew this about me.
I'm a bit torn between "My personal life is none of their business" and "T's need to be represented as well".