Following the definition of a coalgebra found here https://en.wikipedia.org/wiki/Coalgebra, I was wondering if it is a right or a left vector space or both?
Indeed, when we use the Sweedler notation, it seems that it is both. $\epsilon(a_{(1)})a_{(2)}$ uses multiplication on the right and $a_{(1)}\epsilon(a_{(2)})$ uses multiplication on the left. But there is nothing defined on wether it is right or left. Am I getting something wrong?
It doesn't matter. Left vector spaces and right vector spaces are exactly the same thing (just with a difference of notation), so it is harmless to write scalar multiplication on either side. (So, in particular, the left and right multiplciation that are written here are assumed to be the same multiplication; there aren't two different vector space structures.) Note that many coalgebras of interest are also algebras, and so you really can write the multiplication on either side since you can think of it as multiplying by scalar multiples of the identity using the ring structure.