This is really the kind of thing where A/B testing is both necessary and straightforward. Decently intuitive arguments can be made for both sides and there isn't really overwhelming strength on either of them.
A/B testing isn't magic fairy dust that settles design questions. You need to have a clear hypothesis to test. It's especially difficult if the design question concerns long-term perceptions and psychology. An A/B test in this situation is anything but straightforward.
I'd be interested to learn about how this design decision was made in the companies that have had to face it.
* Interestingly enough, it seems like most of the major platforms have moved away from using either "my" or "your" altogether, eg Facebook and Twitter both just have Friends, Followers, Tweets, Photos, etc.