How can you tell if it's safe to hang weight from the ceiling?
It's impossible to say with any confidence without knowing the details of your house's construction.
If there is a regular room above (not an attic) the ceiling you want to hang the rings from, normally those ceiling joists above are also the floor joists of the room above.  If it was constructed to modern guidelines those are going to be pretty strong - if you hang the rings from two parallel joists so the load is spread across two joists, and your attachment is sound and solid, that should be strong enough to hold you.  After all, it would hold you if you went upstairs, stood over the rings, and jumped up and down.
However it may very well cause enough movement to damage the ceiling drywall, make the floor squeaky, etc., especially if you made a habit of it.  If you were jumping up and down on the floor above, the subfloor and floor would help spread the load.  With rings, the forces generated by your weight and motion are focused on a single point and may cause the joists to flex more than intended.
If your attachment is not solid, that's a whole different story.  For example if you anchor your eye bolts into strapping rather than joists, that will either rip down the ceiling, rip out the eye bolts, or both.  If you find a joist but hit it off center and crooked, the eye bolt will look fine from below but will be hanging by a little sliver of wood and will rip out under load.  If you're upside down on the rings when that happens, it will be ugly.