That’s Spock’s tricorder. He would carry extra communicators in case a member of the away team lost one.Just read it this morning. The overhead clamps holding the discs look weirdly uneven on the prototype, but then I guess that's also exacerbated because we've never seen that compartment without the discs!
Looking at Chang's original design: that sure looks to me like it was two communicators being stored in the bottom compartment. (That compartment always reminded me of foldout vehicle ashtrays.) Was it ever specified what they were?
McCoy would carry medical supplies and medicines in his.
Kirk had chocolates and aphrodisiacs.