Can a dog give another dog worms?

Yes, most of the available dewormers do not kill the eggs of parasites or even the parasites (just paralyse them so that these can detach from intestinal wall and pass out with the stool).

The eggs of these parasites can complete the life-cycle and infect the same dog again and other dogs as well.

Thats why it is recommended to give dewormers to all petsin the household at the same time and repeat within 15 days and recheck the stool for any parasites every 6 months or so.