I’ve heard this many times, he was doing something on so-and-so’s dime, ie he was doing it under such a person’s aegis or at such a person’s request (I believe that’s the sense). I presume this must have reference to a phone call, as in to drop a dime on someone, to inform on them.
OED has dimedropper, informant, with a first cite of 1966, but I can’t find the on my dime construction. I presume they’re both roughly of the same vintage, alluding to a time when phone calls cost a dime. Or does that cover too large a period to make that a safe assumption? Does anyone, either with access to RHDAS or greater search powers in OED, know which phrase is the elder?