I observed that when i clicked on my profile name inside community page, it took me to someone else's profile, where i could edit the information. Its a serious security concern. I am not sure if this is temporary system glitch. Please fix it soon.
The reason why it appears like you are logged on with someone else's account is because of a caching issue.
In reality, you are NOT logged on with another person's account. Even if you tried editing a profile, if you would try to save the change it would result in an error and would not be saved. Every data is secured.