Please don't get me wrong. I never said don't talk about yourself and I'm sorry if I gave that impression. By asking people questions about themselves gets a flow going for you to "bounce" off from, and depending on the person, they will do the same. For example, a conversation I had with a coworker today:And no, asking other people to talk about themselves while you don't say anything about yourself is like throwing all the burden on them if let's say they didn't have much to say about themself so you should play it safe, it's also not sharing and connecting and this is not something robotic I'm talking about, I have true emotion and care if I were to talk to someone. And if let's say the conversation ends, it just means that you two weren't connecting well or were taking a break and that's ok.
Coworker: "Hey, if you don't mind me asking, are you married or have any kids?" (Context: I already know he has kids and is married, he mentioned it in a prior conversation, so this question wasn't entirely random nor did I find it inappropriate. Just casual curiosity)
Me: "I'm married, but I don't have kids. I prefer it that way, haha."
Coworker: "Haha, fair enough! I get it!"
Me: "How old are your kids?"
Coworker: "My oldest is 9 and my youngest is 7. I have 2 girls."
Coworker: "Yeah... but they're alot of work, haha!"
Me: "I bet!"
Coworker: "So do you have furbabies since you don't have kids?"
Me: "Yeah I have a German Shepherd and a cat. Plus my husband and I have a small farm too. We have some chickens and goats."
Coworker: "Oh no way! That's cool!"
He then proceeded to ask me a couple more questions about my farm which eventually led to me asking him questions about what he likes to do on his days off, him telling me about working on projects and such, etc. before we went back to focusing on our tasks. He's a very outgoing person, so it was easy to keep a flow of conversation going since he did a lot of the socializing haha.
But my point is, by asking questions to other people it opens up a flow of conversation, thus a level of connection of some sort. It's not meant to place a burden at all and I feel it would only place a burden on someone if you went about it like you were interrogating them, with one question after another with nothing said about yourself.
Also keep in mind that sometimes the flow of conversation won't be that great, especially if talking with other extremely introverted people that don't have much to say. Like you said, sometimes the connection won't be there, and that's perfectly okay.