This is an interesting debate/conversation. On PETA's website, they have a fact sheet devoted to this subject:
https://www.helpinganimals.com/Factsheet/files/FactsheetDisplay.asp?ID=34
Basically, it says that the benefits of a human vegetarian diet can be applied to our companion animals as well, as long as the necessary steps are taken to provide a well-balanced diet for dogs and cats.
The suggestion seems to be to make your own dog and cat food. Here is what PETA says regarding the taurine/L-carnitine needs of dogs: "Additionally, some dogs need two amino acids called L-carnitine and taurine which are not generally added to commercial dog foods and can be insufficient in homemade dog food as well. A deficiency of these nutrients can cause dilated cardiomyopathy, a serious illness in which the heart becomes large and flabby and can no longer function. This illness generally strikes young or middle-aged dogs who are deficient in L-carnitine or taurine because of breed, size, individual genetic make-up, or diet. Supplemental L-carnitine and taurine can be bought at your local health food store"
And regarding cats: "Cats are often more finicky than dogs, and their nutritional requirements are more complicated. Cats need a considerable amount of vitamin A, which they cannot biosynthesize from carotene, as dogs and humans do. Insufficient amounts may cause loss of hearing, as well as problems with skin, bones, and intestinal and reproductive systems. Cats also need taurine. A feline lacking taurine can lose eyesight and could develop cardiomyopathy. Commercial pet food companies often add taurine obtained from mollusks. James Peden found vegetarian sources of both taurine and vitamin A, plus arachidonic acid, another essential feline nutrient. He then developed veterinarian-approved supplements Vegecat™ and Vegekit™ to add to his recipes. These recipes are probably the healthiest way to feed cats a vegan diet at this time."
PETA also lists companies that provide vegan/vegetarian food for companion animals, but they caution you not to rely on this fact sheet alone. There is a book dedicated to the subject,
Vegetarian Cats & Dogs, which can be ordered through (broken link removed)..
Whether or not you choose to persue a vegetarian diet for your companion animals, I think it's important to buy "pet" food from animal-friendly companies, companies that don't test their products on animals (like IAMS does). PETA has a list of companies that don't test their products on animals:
https://www.iamscruelty.com/notTested.asp.