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Well I don't like the looks of that S-trap!
Even with a vent tied in further downstream, it's not a legal plumbing code setup.
Theoretically a good flow can suck that water seal out or partially out and cause gurgling sound. That wouldn't explain gurgling from the other sink, however.

Still more to it.

I would consider adding an AAV after that trap, under the sink, as high as possible,but not so high that you couldn't replace it.