Ultimately this doesn't really affect things. I'm always reaching for the dice to roll but that is because I like to keep things moving. Though I can see how some people may have hangups on somebody rolling the dice when they technically aren't the ones who are supposed to roll.
In the example given it does matter as a timing issue. If I get to roll for the burns on my opponent then I defiantly want the burns to resolve first if they may kill the creature before that creatures regeneration ability gets resolved.