Yes and no. A lot of the sounds are mating calls, but there's so much more being communicated.

Some sounds are warnings, like when squirrels see a cat and start to chirp. If you watch them, you'll see them run up a tree and pause upside down on the trunk, chirping an alert to others. Other squirrels in the area will repeat the behavior and amplify the "message" until the threat (the neighborhood cat) goes away.

Some sounds are intended to trick others. Blue jays mimic the sounds made by birds of prey in order to scare other birds away from their feeding grounds. It works really well - I've seen a jay clear a whole flock of starlings from my yard before. He then swooped down and plucked a bunch of worms and bugs from the soil.

There's probably more, but these are just examples I've observed while hanging out on my porch.