Typically the DM has to do a lot more rolling than the players. So, this can be helpful to some DMs. Plus, even if the players don’t know what the rolls are for, it can make some players feel more involved. And for some rolling dice is simply fun and more rolling means more fun.
You might think that this prevents the DM from hiding the results of rolls, but that isn’t necessarily the case. The DM can ask for rolls without telling the players why, and he can shift the meaning of the results to obscure it further. (Have a handy table for each die type that scrambles the results. e.g. d4: 1=3, 2=1, 3=4, 4=2.) Or the DM can simply still roll for those kinds of rolls.
This can have further effects if you end up changing the mechanic. e.g. When a PC casts a spell, does the player roll a single “reverse save” for all monsters affected or do they roll individual “reverse saves” for each monster? The latter is the same as if the DM had rolled individual saves for each monster, but the former is different. Whether that’s good or bad, however, is subjective.
I suppose the only problem I see is that it takes some effort to carefully think through each reversed mechanic and make sure that it is equivalent or that you are OK with the difference. Plus a bit of effort to translate things on the fly.