Do infants need their own seats on long flights?

The answer depends on the number of people travelling with you and the age of the infant. If you have an extra seat for your infant, you are required to bring your child seat that you use in your car on board. This results in even more baggage. Also you can only be sitting at certain rows, since an additional oxygen mask is required.

If your child is still a baby (< 1 year) and you have at least 2 adults per baby, travelling without an extra seat makes perfect sense. You can shift and babies still sleep in the most uncomfortable situations quite comfortably.

If your child is between 1 and 2, I would recommend buying an extra seat. I am talking from own experience. We didn't in our last journey, because of the hassle with carrying the car seat. We really, really regretted it, afterwards. When our daughters (twins) finally fell asleep the next row decided she would too, and by reclining the seat waking our daughter.

Another source you want to check is seatguru.org. Here you can find the distance between the seats on major airlines. Some airlines still have a decent pitch to travel with a baby on your lap.

This is not something I have done personally but I recently sat on a 13 hour flight next to a family that had 2 children. One was under the age of 2 years old and the other I'd say was 3-4. Our seats were at the very front of the section so we had a little extra leg room and no one sat in front of us. This allowed the family to put blankets down on the floor, and the child laid down on the floor for some portions of the flight when the seat-belt sign was not active. For most of the flight the child slept and was fairly quiet. I'd imagine this was dependant on the child though. Each child I would imagine would handle the situation differently.

UPDATE
I just completed a 13 hour flight (China Airlines) with my wife and 1 year old boy. We hadn't bought him his own seat to save money. Because we were flying with a child we were able to book the very front seats of the section so we had extra leg room. Also because it was the 2 seats on the side of the aircraft, it made it easier because I sat in the aisle and basically blocked our son from "escaping" out of the little area we had made by putting my legs out.

2 things that made the flight much easier were that we brought a full sized bed pillow and a large soft blanket. The pillow was placed on my wife or I's lap and if our son wanted to sleep he could sleep on the pillow instead of our uncomfortable legs. We placed the large soft blanked on the floor where our feet were. This made the little "play" area much nicer to crawl on and some time in the middle of the flight, our son slept on the floor for a good 5 hours. To help him sleep on the floor we draped another blanket (one of the blankets provided by the airline) over top of him to essentially make a little dark fort to sleep in. It worked great.