Wrong: I want to go, I want to stay.
Why: There's a comma between two complete sentences, but no conjunction. (Comma splice!)
Right: I want to go. I want to stay.
Also right: I want to go; I want to stay.
Why: When two complete sentences are separated by a conjunction, they are independent clauses and components of a single sentence. When there is no conjunction, however, they are simply complete sentences, and complete sentences are punctuated with periods, not commas (I want to go. I want to stay).
If you want two complete sentences to be independent clauses but don't want to use a conjunction, the correct punctuation mark is the semicolon (I want to go; I want to stay).