Do I need my own equipment on the field to play Acceptable Losses?

No, you do not need to control your own gear on the field to play Acceptable Losses.

According to RiftJudge FAQ #4344 and #9554, you can play Acceptable Losses even if you do not control any gear, as long as your opponent does. The card does not target, and at resolution, each player who controls gear will choose one to kill.