Space is big and Neutral Zones likely don't extend into infinity. Areas where the Federation may clash with Klingon ships during the 23rd-Century could be in territories claimed by neither side.
As far as violations of the Romulan Neutral Zone (by either the Federation or the Romulans), they tended to be treated like violations of treaty/hostile acts each time. In Star Trek X, though, it seemed as if the Enterprise
was given special diplomatic permission to approach Romulus. I imagine a similar scenario was played out for the ships of both the Klingon and Romulan ambassadors to Earth in Star Treks IV and VI respectively.