Although the sentence "We swing on a swing, and we seesaw on a seesaw" is perfectly correct and makes sense, it has a slightly humorous sound to it because it makes us conscious that the same word is being used as a noun and a verb. This is especially true because we are doing it twice in one sentence, and because the verb "to swing" is common, but the verb "to seesaw" is less common.
It would be more natural to use the words separately. For example:
"The two candidates were seesawing in the opinion polls."
"The children in the playground were playing on the seesaws and the swings."
"My mood has been swinging from pessimism to optimism and back."