@swissprice – There are several “chat” plugins what “work with wordpress / buddypress” – it would be easy to establish a group chat room that allows registered and non-registered users to chat with each other via the main chat room – and they can private message each other there.
As far as “begin chat with this user” – I think you would need something that says “User A” is online now – and a setting that would allow a non-registered, non-logged in person to send a “message”. I think I saw a thread a while back where someone was trying to adapt the “single user profile page” to allow guest comments, aka activity post on that profile / user’s activity page..
Not sure exactly what you are going for here.
Check out Iflychat, it allows non registered visitors to chat, has a couple of different modes and is free for up to 10 simultaneous chatters.
I use Rumbletalk group chat room to do exactly that.
1 – Decide the type of allowed logins to the room.
So if you define GUESTS and Registered users (RumbleTalk), than your Buddypress users will be logged in automatically, guests would provide a name and then they could enter the chat.
2 – Now, users could talk in a group or in private chats.
** They could also have live “one on one” Video Calls or audio calls.
Note, Since they use WebRTC (google video protocol) you will need that your chat will be secured (https/SSL) to make audio and video chats.