Some composers like myself like to have some material to dig into. Visual, narrative, descriptions. Once you have something solid that can direct the composer would start to be a good time. In the film world the composer is usually one of the last people that is brought in. In the alpha stage you can use temporary music to set the mood if you want to. It is a testing phase anyway, For the beta I feel it would be appropriate to have the music finished since it is not only the production team that tests the game.
Regarding the free music/paid music part... Once the composer starts to get more serious about his/her work they like to get rewarded for the time they give. Especially if they have professional goals. I understand it is a dilemma when your game is going to be free to get and play, Some paid composers are good. Some free can surprise you, In the end it comes down to where you draw the line with your own goals and ambitions.