The answer to your question is "both" (sort of). The reservation is in your name, but under the travel agent's control.
I'll assume for a minute that the agent placed a "courtesy hold" on it, as opposed to having you put down a deposit.
In that case you just have to let the courtesy hold expire, and the other agent watch for it to be released, and then grab it.
If you placed a deposit, you'd have to have the agent you booked with release the booking to you to transfer to the other agent.
No doubt if you asked the first agent if they follow the booking for price drops they'd say yes.
But.. honestly, I don't believe any agent will check your booking daily or twice daily for price drops. Everyone should be watching for price drops themselves, as it's very easy to do online, and good agents are too busy to babysit all their reservation that closely.