I hate to disagree twice in a thread, but...
Modern "moly" rings are low tension and assuming the machine shop followed the ring manufacturer's comments for installation regarding hone job and clearance, they are broken in with the first dyno run or first romp on the street.
A long break in period was true back in the '60's. Today, power increase is almost imperceptible with time on a properly assembled engine and coil consumption should be about as good as it's going to be.
If you use a machine shop that does not do it the way the ring company suggests, then you are at the wrong machine shop, imo
Mass production rebuilders may be another question