Sounds like you have a couple things to deal with.

Normally one would check all the connections, then plunge the sink, etc. But since it's already getting wet under there, first have one person watch with a flashlight inside the cabinet as the second person operates the disposer. Find the leak.

It could be coming from the flange connecting the disposer mount to the sink.
or the disposer itself (eg. Insinkerator Badger's can eventually rust and leak right out of the bottom)
or the disposer elbow
or the P-trap, etc.

Then fix the leak (tighten or repair).

Next, plunge your sinks. Hopefully that will take care of it. To do that, use a wet rag to seal one drain as you vigorously plunge the other drain (with some water in the sink). Might want to have one person hold the wet rag as the other does the plunging (don't use a plunger that's been in the toilet - have a separate one for your sinks.)

Good luck.

This post was edited by homebound on Thu, May 23, 13 at 9:52

If water can run for 5 minutes before the sink backs up, there's a blockage or drainage problem of some kind in the house's sanitary sewage system, far away from these sinks. Call a plumber.

When he comes, he can also fix the under-sink leak, that's probably much easier to deal with by comparison. Don't be surprised if you need a new disposal.

Thank you both for your input.

I live in a condo (ground level) and this has happened 3 times - 2009, 2011 & 2013. It tends to happen around spring. The complex is 12 years old with one medium size tree 15 feet away (tree roots interfering?). The same event occurs every time. Water backs up and leaks when the compactor is on. When it is not clogged, there is no leak when the compactor is on. If the clog is far away, is it possible that my neighbors are experiencing the same problem (clog)? I assume all our pipes are connected at some point. I am tired of having to deal with this every year or every other year. I live alone and rarely cook so it is not like I am flashing down a lot of grease.