Yes, the rice sized worms you saw probably were the egg sacks of tapeworms. And most importantly, yes, humans can have them too -- but obviously not as often -- we are not the natural host. (This is why I do not let dogs or cats lick my face -- they lick their hinnys because they itch and then lick my face? I DON'T THINK SO!)
I don't know if doctors would treat for these unless the child was showing symptoms. Lots of pets get them (fleas are the carrier) and I haven't heard of children being treated -- but it may fall into same category as "dirty"?.
Please don't confuse these worms with the pinworms (and dogs are not hosts for these) that a lot of children get in preschool or elementary school (which is probably what you had as a child). Those symptoms are never sitting still -- wiggling around on chairs -- and the nose always itches. Pinworms come out at night around the anal openig when your child is sleeping (if the child has them bad enough you can see them when they poop). You can go into their bedroom, spread their "cheeks", and see the worms (use a flashlight!). Doctors usually send moms home with what the lab calls a scotch tape test....
When you shake the sheets in the morning, not realizing your child has the pinworms, the eggs fly all over and then, you too, can get them. BEEN THERE, DONE THAT! Very contagious...
You breathe the eggs in, they find their way to the intestine.
Well, more than you wanted to know. Perhaps one of the other mothers has had actual experience with "dog" worms.
I just read some of the other responses to your question.
Pinworms are not the same as tapeworms, although the outcome (rice) looks the same. I could give you the Latin names and life cycles .... and yes, one mother was correct in saying that if your child runs around barefoot, you could get worms -- those would be hookworms. But the yard has to be very infested, conditions just right, etc. Lots of things "out there" that just love us -- but not much stuff gets us, thank goodness!