According to some authorities, such as Purdue OWL, yes, a comma should be used before "and" in that sentence:
Use commas to separate independent clauses when they are joined by any of these seven coordinating conjunctions: and, but, for, or, nor, so, yet.
It's a compound sentence, as you identified, and should therefore have a comma.
That being said, a more reasonable guideline, as given at Grammartips.homestead.com, governing comma use is that they, like all punctuation, should be used to reduce or eliminate ambiguity. You can often eliminate the comma
if both independent clauses are quite short, especially if the two clauses are very closely related, and even more so if the subject of both clauses is the same, or
if only the first clause is quite short, especially if the two clauses are very closely related, and even more so if the subject of both clauses is the same.
Here is an example involving two short clauses conjoined with 'but':
John went to the store but he didn't buy anything.