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 yourself, 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:
I don't remember his name / I don't recall his name. → No recuerdo su nombre.
Remind me of his name / Recall to me his name. → Recuérdeme su nombre.
In both cases, the words recordar and recall are more about the act of pulling something from memory (similar to remember), 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 traditional English ballad made famous by Simon & Garfunkel as Parsley, Sage, Rosemary and Thyme:
Are you going to Scarborough Fair?
Parsley, sage, rosemary, and thyme
Remember me to one who lives there
She once was a true love of mine
When Spanish people learn English we have the same problem, why do you use two verbs ? For us
recordar is used in both cases.
Traer a la memoria algo.
Whether you want to share it with someone or not is up to you, because before reminding somebody to do something, you have to remember it ;).