I don't think it's the T or E for me. It has fluctuated and changed over the years, but appears to be independent of hormones and have more to do with how isolated I am, how sad/happy I generally am (in terms of mental issues or lack thereof), what people I'm around, other life circumstances, etc.
I have a tendency to withdraw when I'm upset or moody, and not want or enjoy talking to people then unless they have some good practical solutions maybe. I hate trying to explain what I cannot, to only get sympathy in return. So venting or seeking emotional support has never really been my thing. Sometimes I rant, but that's more just to get stuff off my chest and not to get a response.
However when I've been in more happy/joyful/stable periods of my life, I've also been more outgoing and enjoyed socialising. I have a very black and white perspective on being social: either I want complete solitude for months at an end - or I want constant attention, and there's no in between.
Those two moods/states have switched the same whether I've been on or off T, so I think that's just how I am. But your question is interesting nonetheless, and I liked playing with that thought.