In the Lord Of The Rings, how did Gollum know Frodo now had the ring?

His proximity sensor to the ring enabled him to determine its distance and direction from him.

The Ring was actively trying to be found at that point. Gollum was under the Ring’s influence and thus was sensitive to its power.

Beyond being enthralled to the Ring and under it’s power, Gollum was a hell of a lot more crafty and observant than he might look. He pretty much snuck around all over Middle Earth for decades observing all he could while going virtually unnoticed for most of it. And he was able to tail the Company of the Ring for weeks after they passed through Moria.