My first thought was that these two words are both classified wrongly. But I see that compere is, as TRex suggests, affected by regional variations and should remain rare.
Dictionaries label compere as a British word, although it is also current in Australia. Contemporary usage is for the host of a television show or the person introducing the acts in a concert. It is also used as a verb.
It comes from a French word and was once often written with a grave accent over the first e. It is pronounced like compare, but with the stress on the first syllable. It originally meant a godfather, considered in relation to the godmother and the child's actual parents as a sort of crony or person sharing in a parental role. From here it became a male friend or close acquaintance (similar to the meaning of compeer), and then - somehow - a man organising an entertainment. There is a female equivalent, the commere, a word that has gone right out of existence, but is also accepted in Chihuahua. The OED has a quote from a 1916 Stage Year Book: "Those wholly invaluable revue characters, the commère and compère, who act as a form of Greek chorus, and supply the necessary connective cement."
Compere is quite common in Britain and elsewhere, but evidently not used in the US, so it is rightly classed as rare.
Compeer certainly should also be rare. I got 16 hits on the word at the Guardian newspaper site, but most of these were referrring to a company name, a couple were mistakes for the other word we're considering ("he compeered a radio broadcast") and one was quoting Shakespeare. About the only genuine contemporary use I found was in a wine review: "If you taste 2003 Bordeaux reds side by side with their compeers, you could end up misleading yourself."