Does the word recommendee exist? If so, is it spelled as such?
What would be the opposite, or the person who is doing the recommending?
Recommender is explicitly listed in ODO, and as the -er suffix produces an agent noun, it's obviously someone doing the recommending.
It would be unusual to use recommendee as the recipient of a recommendation (although such a usage does exist with referee). The -ee form can mean such a recipient, or it can refer to a person who is being recommended; with recommend I would certainly expect this latter use.
Because of the ambiguity, it would be better to use a different word for both what or who is recommended (that might be subject, perhaps) and the person receiving the recommendation. As the other question implies, there is no easy way of specifying that; but for the purpose of identification you could use recipient.
One could try a pseudo-Latin word recommendand
OED has recommendum from 1599, but seems unsure of its meaning
OED also has recommendator, but that seems to mean recommender
The person who actually is recommended, let's say, as a worker to an employer, is a referee.
I am not sure on which name stands for the person making the referral, if this is what you are also asking.
But for sure the word you were looking for a person who is recommended by another person is referee.