Last weekend in an NSA tournament, PU (appropriate acronym) calls an IP. HC questions what was Illegal about it. ump said that (based on her follow thru/finish) she was releasing the ball above her hip, which is against NSA rules. HC explained that if she had been releasing above her hip, the pitch would've been over the batters head. "Read the rules. the way she pitched is not allowed in NSA".
I spoke with another ump in the parking lot later and he had no idea what the guy was talking about, why he was calling an IP without any coach bringing it to his attention first, and why was the PU calling IPs (as opposed to the BU)
Without seeing the pitch have no idea what the pitcher may have been doing, but the rules do require the hand to be below the hip. I have seen pitchers that do short arm the ball and their hand is above the hip at release, this may be what the umpire was indicating but again have no idea without seeing the pitcher.
As for calling it, no umpire should ever wait for a coach to complain about an illegal pitch before calling it. If the pitcher is illegal it should be called immediately, whatever umpire said that they should wait for the coach to bring it to their attention is not doing their job. And either umpire can call illegal pitches, it is not just one or the others responsibility. There are some differences in which umpire is looking for what, the plate ump is responsible for the hands, double touch, lane violations etc and the base umpire is responsible primarily for the feet and watching for number of arm rotations.