So, if I have understood correctly, when I send someone some bitcoins using the client, I pay some of them as a fee?
Possibly. Depends on the factors mentioned above (see the link I posted). The fees you are talking about are charged by exchanges when you convert bitcoins to dollars and vice versa. These fees are used by the network to avoid spam transactions.
If you are using a fountain service, I would advise pointing your funds to a balance (not necessarily a wallet), so you are not stuck sending the coins you received from the fountain.
Of course, most exchanges/wallets/accounts charge a fee anyways, but this should still save you time and/or money.