The simple answer is NO, there is no problem.

Thanks. For some reason, my husband thinks it's hard on them to have water running through them all of the time. I still doubt he will admit he is wrong though.

Ah contrare mi capitan,,Not only is that excess water not harmful, it is actually very good for it. When vegetable matter is ground up in a disposal it ends up in fine pieces about the same consistency as cole slaw.

The leading cause of failure in garbage disposals is corrosion of the pot metal housing which results from acids that are produced by bits of decaying vegetable matter clinging to the inner wall of the housing. The more often we run water through it the more efficiently those particles are washed down the drain, thus reducing the corrosive effects of the leftover foodstuffs and vegetable matter.

To get maximum service life out of a disposal you should get in the habit of running the disposal about ten seconds every time you empty a sink full of water or at least once a day even if you don't grind food waste, especially after washing pots and pans.

About once a week you should drop a couple handfuls of ice cubes in the disposal, turn the water on, then run the disposal until the ice cubes are gone. Grinding the ice cubes will help clean the grease and food matter buildup off the cutting edges of the rotating cutter wheel.

having two basins and one garbage disposer is common.
And I like to run water for a while to move things down the line.
For a while, the City of Seattle was handing out restricted aerators for kitchen faucets, but then found that kitchen drains were plugging up.
Water is needed for them to run clear.

Here is a link that might be useful:

"The leading cause of failure in garbage disposals is corrosion of the pot metal housing...

This the reason higher quality GDs use stainless steel for the grinding chamber, slingers, etc.

If you want a GD to last get at least a 3/4 horsepower unit with a stainless steel grinding chamber.

The Insinkerator Evolution series is what you want. Yes, it's a bit pricy, but it won't rust out and leave you with a cabinet full of water damage. It'll also be very quiet and give you a better quality grind if you choose the one that has a 2 stage grind.

Thank you all! I actually do not use the disposal that much as I compost most of my kitchen waste. We've lived here 25 years and I think we're on the second disposal. I just needed to convince DH that we really could get by with a single sink. I will try to do a little better daily maintenance though.