You can customize numerous aspects of SwiftKey Keyboard, but keypress sounds have not thus far been one of them. As of now, they are. The latest update to SwiftKey includes the option to set one of four keypress sound profiles. There's Traditional, Android, Modern, and Blip.
The keypress sounds are available in the keyboard settings under "Typing." Modern sounds like a woodblock, Android replicates the default Android keypress sound, Traditional sounds like a clicky mechanical switch, and blip is a little electronic beep. The sounds are still off by default, which is fine with me.
It's unclear if there are other changes in this update, but the keypress sounds are all SwiftKey opted to mention in its blog post. This update is still rolling out, but you can install the current beta if you're weirdly anxious to test the new sounds.