It really depended on which Trade Union was in charge. In the CNT, for example, the militias democratically elected leaders, but their authority was temporary. It might be best to think of it like a temporary task force commander who can be replaced after the conclusion of an operation, though even this comparison isn't entirely accurate.

In short, there wasn't any typical signs of military professionalism. Leaders as said were elected, so you had commanders and fighters, more or less. Orders were derived, theoretically, from the Republican Ministry of War but in practice were mostly disregarded. The Unions and local committees more or less directed the militias, though even this was tenuous. They often operated fairly autonomously.

This all eventually ended around 1938 when it became clear that the Republic was losing. Many militias were folded, after much negotiating, into a moderately unified Republican Army though still the militias operated in a fairly autonomous manner regardless. They just paid a little more attention to the directives from higher authority.

Edit: It should be noted that not ALL militias folded into the Republican army, just the majority of them. Some still refused to be part of any hierarchy.