向こう is relatively abstract in its usage and can include both the literal sense of being "on the other side," like in bridging some sort of gap, as well as in a metaphorical sense of "over there," like referring to people, as in me and the other person/people/whoever. 向こう also carries this connotation of general "awayness" that you can use to refer to a place as vague as "over there" or another country.
向こう側, on the other hand, is a little bit more limited in scope because you attach the extra 側 meaning to it. What 側 does in this sense is it kind of makes it a little more concrete, if that makes sense. Basically in every usage that I know of for 向こう側 you could say 向こう, but for every use of 向こう you cannot say 向こう側.
For example, 向こう側 refers to the other side of something. So you can have 川の向こう側 or 山の向こう側 to refer to the other side of a river or the other side of a mountain. In a conversation you can have your point of view and the other side's point of view, and you can also use 向こう側 here. The nuance here is that there is an emphasis on some sort of spatial/metaphorical other side. 向こう by itself, I believe, has more of a general meaning of away. There is the related word 向かう which means "to face," and its noun counterpart 向かい, which is a synonym for 向こう. Thinking of it in terms of facing something/somewhere may help you understand the scope of 向こう in general while 向こう側 sort of anchors that down to just another side.
If this was a little long and rambly then I apologize, and I'll try to boil it down to a few basic points.
- 向こう generally means the opposite direction or something located there, like 向こうの返事 or 向こうの家
- 向こう can include location that is generally just away from the speaker's location, especially with regard to another group/individual or whatever, like saying 向こうの人 when referring to a foreign country in relation to your own.
- 向こう側 basically means "other side" or the "far side" or the "opposite side" or whatever, as in the far side of a mountain, the other side of a river, especially as contrasted with this side.
- Both can refer to another party in some conversation or in events (相手) but generally I think 向こう is used more.
The following links are useful for seeing the distinctions (in Japanese):
ALC entry for 向こう
ALC entry for 向こう側
Dictionary entry for 向こう
Dictionary entry for 向こう側