Latest data shows that about 29% of U.S. banknotes in circulation are $1's (12.4 billion out of 43.4 billion). Very few $1's, or other low denominations, end up overseas; that happens mostly to $100's. We're very happy to provide currency to other nations; what we get in return is an interest-free loan of something close to a trillion dollars. As for purchasing power, granted that $1 now has the purchasing power that a dime used to, but why does that imply that it ought to be a coin? I'd be more inclined to conclude that, since people clearly prefer $1 notes to $1 coins for most purposes, maybe notes are just more convenient than coins. The only reason the dime of your childhood was a coin instead of a note was that it was silver, and people preferred silver to paper (until the denominations got high enough that the silver got too heavy). Now that our coins are base metal, we might as well replace them all with notes--or wait until inflation does that job for us! (Not entirely tongue-in-cheek, by the way--I'd really like to see a 25c or 20c paper note issued. It might very well succeed in displacing the quarter, in the way that the $1 coin has largely failed to displace the $1 note.)
There's very little I found on the net about US one-dollar bills [$1] circulating abroad, especially in local markets... just the Zimbabwe bond coins 1c.-50c. which imply local $1 note use. What about counterfeit $1... ? I guess that's their problem! Maybe the Bureau ought to put The American Flag on $1 notes... even for countries that won't pay seigneurage seignorage [CT spell check doesn't like either one] ...so our Proud National Banner can grace local junk food transactions ...around the World!