I can't speak about felines, and I'm not sure if doggy-related information would be relevant here, but my dog, Butters, had tapeworms when she was a young pup.
She showed NO symptoms whatsoever (beyond pooping out little bits every now and again). The vet wanted to confirm it with a fecal sample, but of course, tapeworms only show up in the poop during certain times. Which means you either have to get lucky, or try to conserve a tapeworm sample from home.
Anyways, our vet was nice and took my word for it (since I KNEW it was a tapeworm), and gave us 2 tablets of "Virbantel" - a dewormer. I believe they have a cat equivalent available under the name of "Drontal", although it requires a prescription.
The pills worked well and she hasn't had a single worm-related issue since.
Proud parent of Scamper (age 12) and Robin (age 4)
At the rainbow bridge: Slippy, Faramir, Gabriel, Lucky, Ammy