Question Market-maker vs Agent (direct trading)?

Currently I’m choosing a forex broker. This morning the following thought is bothering me. Some of the brokers are market-makers (they’re hedging), some of them are agents (directly on the fx market).
Those that are hedging are actually taking the opposite position I took, i.e. if I sell EUR/USD, they buy EUR/USD. Am I getting it right??? And they bare the risk.
(In case I understood correctly the hedging idea for the market-makers) let’s assume that the Euro continues to fall for a long period. Of course the traders will short EUR, in this case the market-maker is buying EUR. In this case are the market-makers loosing money while their customers are winning???
(In case I’m right of the above) and if the EUR continues to fall are those brokers good/secure enough to sign up with them?
Thank you, guys. I’ll be happy to hear your thoughts on this.

Don’t worry too much about the why and wherefores, the top brokers execute your trades well whatever way they operate, Oanda is one, there are others but I can’t vouch for any others.

Thanks purplepatchforex! Thanks to the search in this forum I found very good, detailed explanation about the market-maker and I’ve realized that previously I have not quite understood the concept behind “market maker”. In this forum I really find very useful information.
Thank you!