[subject] + 一边 + [verb phrase] + 一边 + [verb phrase]
[subject] + 边 + [verb] + 边 + [verb]
一边…一边 puts two actions side by side: the same person does both at the same time. Each 一边 goes straight before its verb, and the subject comes before the first one.
Each half is something a person does, and does over a stretch of time. Two adjectives don't fit; they take 又…又.
The pattern needs a single subject. Two people doing different things at the same time take 的时候.
With one-syllable verbs, 一 can drop out: 边吃边聊, 边走边说. It's short and common in speech.