I ask because I have a small raised bed wrapped in hardware cloth to keep the wildlife out, and then it's covered in shade cloth to keep the hot southwestern sun out. The plants (one "container tomato" and some herbs) themselves have thrived since I put the shade cloth on, but there is limited access to any bees or other pollinators. I've seen new tomatoes on the plant, but I'm wondering if bees had more access, would I see more fruits?
For the most part, no, tomatoes don't require pollinators. Because of the way tomato flowers are formed it is very common for pollen to fall off of the stamen (male-part) onto the stigma (female part), achieving pollination without any outside intervention. In some varieties with particularly short stigma, the flowers are often pollinated before they even open.
The reason I say "for the most part" is because the flowers do need some minor amount of disturbance to shake the pollen down. This could be from a bee stopping by or from a person nudging the flower gently, or even from a light breeze passing through the garden.
They are self pollinating, but as the first answer suggests, do better with some help from air vibration at the least to get the pollen out and in the right place. The link below makes an interesting read and explains exactly what the situation is
Yes, tomatoes require pollination. If you are seeing fruit on the plants, at least some pollinators are finding their way through the obstacles. If you notice very little fruit set for the number of flowers you are seeing, then you may want to loosen up the defences a bit.
As the other answerers have stated, tomatoes are self-pollinating. However, it should be noted that tomatoes can be pollinated by other tomato plants (hence accidental crosses, with cross-pollination). Therefore, it stands to reason that having multiple plants may increase your odds of more thorough pollination. More thorough pollination doesn't necessarily mean more fruits, however, because each unit of pollin corresponds to only one seed of a fruit. However, I would suppose it's possible that you may get more fruits as a result, but I have no idea how much more, if true. My guess is you might get an occasional extra tomato.