You can make it so only members of specific usergroups can post images; how those users get into the usergroups is up to you - if you integrate with vBulletin, then it can be done automatically using the Subscription option, if not then its a manual process.
We use the term gallery for public galleries anyone can post in and albums as a personal gallery that only the owner can post into.
Users can view all of their own images grouped together by going to their "Member Gallery" which is a psudo-gallery made up of all the users posts.
Based on what you are asking, I see no reason PhotoPost cannot accomplish what you want.
Please do not PM me for support or sales questions. Thank you for your understanding.