As far as I can see, there isn't any. Bottle caps are accepted by all merchants, regardless of where you are, so there isn't any need to have Legion coins to trade with the Legion. You don't even seem to get a favourable deal if you trade with the Legion using their own money.

Am I missing something?

For both: Loot. I take them as spoils and then use them as stock for buying casino chips. That way, my caps aren't touched and I ultimately don't actually lose caps if I get an unlucky streak. When using Legion Denarii as that, it's a humorous irony: Legion money being used for acts of profligacy.

For Legion coins in particular: Load them up as 12 gauge coinshot to get Buckshot Magnum power without the extra wear and tear. If you're on PC, you can use one of my mods, http://www.nexusmods.com/newvegas/mods/43420/?, http://www.youtube.com/watch?v=FMUOZSxiNJc.

NCR dollars... Idunno, toilet paper? They're devalued after the Bros irradiated their bullion.

Well they cant trade with bottle caps forever so sooner or later youd expect them to introduce some sort of currency that doesnt involve caps. In NV its early stages still so caps are still the most prevalent currency. May not suit the game as you say but fits the logic as to why theyve started introducing it.

Creates artificial demand which therein gives the currency value. Although the idea isn't quite explored in-game, look at it from the following perspective: the NCR introduces paper money to counteract caps for markets in their sphere of influence. Hence, caps are now valued are more than just a one-to-one ratio. Similarly, the Legion introduces their currency primarily for caravans they trade with, providing "monetary" value for goods and services between them and traveling merchants. Wherever you introduce services and currency, you automatically create value where it may not have existed.

I always assumed the currencies were a viable alternative but never used because House controlled their supply and demand through caps.