If you want to attract someone's attention to a comment of yours in a thread that's no longer active, you can page them using the syntax
@fedja in a comment, without the need to pollute other threads. It will show up as a very visible notification on the upper left corner the next time they log in. If the user has email notifications enabled, they'll eventually get an email saying they have pending notifications in their inbox.
After establishing contact, you can interact in that comment thread, move to chat, or if you want more privacy move the discussion to email. If you want to initiate an email discussion without making your email public, once you have a live conversation on comments or chat, and if they agree to do so, you can provide your email in a comment and delete it after they have seen it.
Private messaging has been consistently declined across the Stack Exchange network, for a number of reasons:
The consistent response to this is that it's a bad idea, because:
It could hide information from the community: useful information transmitted privately is unavailable to other readers, subverting the core purpose of the site.
It could be used to harass other users ("Answer my question!", "Accept my answer!", "Yer momma so fat she overflows the stack!", etc...)
However, there is some value in being able to communicate with other users outside the context of a specific question or answer. This point was conceded with the implementation of a chat system: users on every Stack Exchange site can create and participate in chat rooms, integrated with the normal user accounts. There are still no truly private messages between users, but for those who want to chew the fat or discuss whatever outside the confines of the sites themselves, this can provide a viable solution.
For normal users, the advice remains: if you want others to be able to contact you privately, add basic contact info to your profile's bio page.
While a private messaging system appears particularly helpful to Math Overflow - which is not as essentially a Q&A site as SO, and a bit more of a place to generally discuss research-level mathematics - I think it's not as useful as one might think. MO probably does have a lot more interaction and gives rise to a lot more off-site collaboration than other SE sites, but I don't see a strong incentive to keep it off the public record provided by the chat. If you want to collaborate on something that will give rise to a paper and keep some details private, you'll probably want to move to email soon anyway. If a discussion is getting ugly enough that you want it off the record, you should flag a moderator as soon as possible. (One course of action is then to have a moderated discussion to see if it can be resolved, and the moderator can later delete the chat.) If you just want to chew the fat in private, MO is probably not the place.
[ chat ]to get that link $\endgroup$