The flash freezes the action. Typically with flash you'll be shooting at 5.6, 8, 11 etc which means at 1/60th you'd have to have something extremely bright to have any affect of the photo. If you had all the room lights on in a house and shot with flash at 1/60th, the room lights would have no effect at all at fstops like f8, for example. By the way, 1/60th was the standard sync speed for most cameras when I started photography and there was never any problem. I still shoot at 1/60th usually out of force of habit.
Of course, its not totally clear what you are saying. Did you pop up a flash on the camera itself and fire that to cause another strobe to fire or are you saying that there is something wireless in the camera that fired the strobe? If you did have an on camera flash that fired the strobe, then it also will contribute light to the subject (and in that case, you'd usually point it at the ceiling or deflect it with tinfoil, etc.