I used Google to translate "remember" and "remind". Both came out to be
recordar. Why is there no distinction made? You can only remember something by youself, but you have to remind someone else of something. How can recordar represent two different actions?
Recordar (and incidentally acordar) in Spanish works more like recall in English, than like remember or remind. In English, recall can be used in both senses as recordar, and context determines the meaning:
In both cases, the words recordar and recall are more about the act of pulling something from memory (similar to remeber), than the act of communication (as in remind). In the case of recalling to someone, it's the "to someone" part that implies the communication, not the verb itself.
EDIT Remember in English can also work this way, although it's rather rare, archaic or used for artistic purposes. Consider the Simon & Garfunkel lyrics from Parsley, Sage, Rosemary and Thyme:
When Spanish people learn English we have the same problem, why do you use two verbs ? For us
Later if you want to share it with someone or no (by yourself) it is up to you, because before remind to somebody to do something you have to remember it ;).