Ah, thanks, that was my confusion; I thought the ecash was the token being stored on the device. Didn't realize ecash was an IOU held by the mint operator. The way it's presented on the website is: "Ecash is a digital barer token that is stored on a user's device, very similar to physical cash."
https://docs.cashu.space/#
Yes token is on device but is only redeemable for actual lightning sats by the mint that issued it
If Alice has tokens issued from Minibits and sends them to Bob who uses enuts, can he redeem the tokens from Alice? He would be trying to cash in tokens that enuts did not issue.
Yes because Alice's ecash token would be redeemed for sats that would be paid over lightning network from minibits ln node to enuts ln node which would then issue Bob ecash tokens that are kept on his device
If Alice and Bob share the same mint then they can send ecash to eachother without a lightning transaction and with near perfect privacy
iiuc, Alice & Bob can send the bearer tokens amongst eachother intra mint with no Lightning tx but to send inter mint; the token has to be redeemed by the Alice's mint, sats sent via Lightning tx to Bob's mint, and there new bearer tokens are issued to Bob.