I've found that the soaping action is maximized if you give the sponge some time to absorb water before you start scrubbing.
With the car sponge, I toss into a bucket of water for about 5 minutes while I'm hooking up the hose, getting the car into position, and hosing it down. The sponge grows much larger and softer during that time, and then there are plenty of suds when you squeeze it during washing, especially if you dip it back into the bucket every so often. I don't find that the surface water on the car is quite enough to get the best performance out of the sponge.
Same with the pet sponge. Let it soak and absorb water for 5 minutes while you're gathering towels and adjusting the water temperature. If you do that, and remember to keep squeezing as you scrub, there are plenty of suds.