My bathroom sink clogs up when I run water from the kitchen sink.

Might be your vent isn't working. Could be a grease or hair clog where the lines connect.

your vents are probably clog. In this case you need to get on the roof or in the attic and ream them out. years of steam carries grease up into the pipes and cause them to clog. You bathroom and kitchen are most likely on the same vent. This is not a drain problem

That definitely sounds like a job for a to me. It seems like there is something stuck between the two places. The only other thing to try is to snake the drain. Your local big box store probably rents these.

I wish I were home so I could tell you the name of the product that's MUCH better than Drano. It's sold at Home Depot, Lowes & Walmart. At Home Depot & Lowe's the bottle is in a plastic bag, maybe because it's bigger than the one I saw at Walmart, not sure the reason. (I think it's Instant Power Grease and Hair Remover) It cost more than Drano but is well worth it. I hope you find it and good luck.

Try using a plunger in one sink then run some water. If it's still messed up use the plunger in the other sink. If that doesn't work call a plumber.

I would call Roto Router, you may have a clogged sewer line, roots most likely.