A quick survey of the internet reveals many instances of "home sweet home", no punctuation. But doesn't "home, sweet home" make the most sense?
Yes, it makes the most sense. But, for one reason or another, the written version doesn't tend to bother with any punctuation at all. The closest thing I can find to a definitive answer comes from the apparent source of this phrase. I don't know for certain, but I think it is extremely likely that this common phrase came from the song of the same name. As you can see in the link, the punctuation in the song's title is either "Home! Sweet home!" or "Home, sweet home". So, the way you want to parse the phrase at least agrees with the creator's intent.