I have 3 objects. I select all, and would like to rotate each around the Z axis of each object (locally) not global.
I need to rotate each object about its local axis.
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Now when you rotate, either by dragging the blue handle of the rotation manipulator, or by pressing R ZZ it will work like you want.
There's a button to pick transformation mode, just above the timeline in default view. It will be showing two orbs overlapping with a blue mark in the middle.
Click that and select the "Individual origins" setting (two orbs with individual blue marks).
Now every object will rotate around its own origin point.