well the current would be dictated by the impedance of your load.
V=IZ
if youve got 8Ω speakers, then I = V / 8Ω
what are you setting your voltage to? if its not high enough then your speakers probably aren't getting enough power.
as far as I understand, the black rings aren't connections, just insulators. unless you have specific shielding connections, in which case its not a standard TRS. if you have separate shield connections for each line, then ground them all. if that doesn't work then find a manual on your type of connector and find out what the shields should be.
if they are stereo headphones, the longest connection on the plug (on the shaft, farthest from the tip) should be shield, and this should be grounded. the signal may be applied to either of the other two connections, one being left and one being right.
my guess is power though, so try increasing your voltage a bit, make sure you dont blow up your signal generator either, as some speakers can be just 2Ω... also, your signal generator and your ground connection should share the same ground.