Is private chat possible?

Could someone tell me in which of the instances private chat box is available ?

To my knowledge there is only direct messaging between users; but it isn’t end to end encrypted. The admin(s) of the instance can see the contents of the message.

So its like e-mail, but not chat , is it ?

Essentially yes, it does not appear to be posted publicly, but it is unclear how or even if this could occur between people on different instances given the way ActivityPub, ActivityStreams and OStatus work. Without separate precautions it will, at the very least be visible to instance admins and possibly moderators.

If private messages are solely facilitated by not publishing a link that’s otherwise accessible (I haven’t checked and I can’t touch Mastodon code due to the license, especially not for security related code) then it’s most likely simply a matter of time before someone gets caught out by that.