In several occasions, for example, when Mikami used his Shinigami Eyes to see everyone's name in the final episode, he saw in a red tint. Is this only done for dramatic effect, or do they generally see in a red tint?
It looks like they do see in a different colour. The first time Misa meets Light we can see her look at him differently, but from the manga it is hard to deduce what the actual colour is. From chapter 28 it does seem that users with Shinigami Eyes do see in a different colour as shown in the pictures below. Another moment when this happens is when Chief Yagami looks at Mello through the Shinigami Eyes. At that time, he also sees him in a different colour, so it is safe to say that people with the Shinigami Eyes look at the world differently than we do, but the actual colour could be any colour, since the manga is just black and white.
In chapter 28, we can see this colour page and it would make it seem that users with the Shinigami Eyes, see the world not in red, but in reverse. Although it is not completely sure who made this colour page. It might just be fan-art.