Can each player fight a minion in each round in Roll Player - Monsters and Minions?
Yes.

On page 6 are the rules. The important bits are as follows:


Examine the face-up top card of the Minion deck. (If this is the first turn, this card will be the one pulled during setup.)


...


If the reward level does not have the trophy symbol, put the Minion card on the bottom of the Minion deck, because the Minion survived and ran away. (You did not defeat the monster, bottom deck it.)
If the reward level does include a trophy symbol, place the Minion card near the player’s Character Sheet as a Trophy: the player’s character has killed the Minion! Then gather information:... (You did defeat the monster, if it's a trophy keep it.)
Reveal the top card of the Minion deck and place it face up on top of the deck. (Draw a new monster.)


Regardless if a monster is defeated or not step 9 ensures that a new monster is placed. All the steps on page 6 occur when the current player goes on a hunt, not once per round. In short, the monster defeated or not gets replaced by a new monster that is available to fight for the next player in turn.

You can download the rules PDF on BGG