No, we shouldn't.
Stack Exchange isn't a social network; it's a questions and answers network.
Now, if I was an expert in geology with specialization toward volcanoes, and saw multiple questions tagged volcanism in a short period of time, I might get curious and look more closely at that tag thinking that maybe there is more there which I can answer relatively easily.
Same with any other subject.
But we shouldn't expect people to do that. Every user picks which questions they want to answer, and to what level they want to take their answer, based on their own feelings, available time, and knowledge. I know I have been rather annoyed when people have taken my answering one of their questions as some sort of carte blanche to ask me specifically to answer their other questions as well.
Also, for your specific suggestion of chat: generally speaking, chat is a terrible way of doing Q&A. It comes down to largely the same issues that make the Stack Exchange Q&A format better than forums (or Facebook) for getting specific answers to specific questions. If someone is willing to discuss a question through chat then by all means, but those discussions should focus on asking the best question possible. The answer should be posted on the question itself. That allows the community to chime in and vote, potentially much later, and preserves knowledge in an easy-to-find form for later.
(I have started out in chat a few times trying to figure out basically where to start asking a question, and what information should go into the question, when I wasn't sure how to approach a problem. But I have always made it clear up front that the intention is to post a question rather than seeking actual answers in chat. That also means that people not intimitely familiar with the subject matter might be able to help out in chat, as asking a good question is a skill that transfers between subject matter domains.)