House managers are the equivalent of prosecutors in a trial. Remember this is not a true legal trial.
President picks his own choices to be the defense.
To the question about about R "managers". In theory (though not reality) this is not a party thing. This is the House accusing then trying the case with the Senate acting as jury and the SC justice as the judge.
In reality the House sent all D's, which is their right, but destroys the "non partisan" aspect that is supposed to be in place for an impeachment.
If, and that's a big IF at this point, there are witnesses, both sides get to call them and question them. It is still unknown how this will play out, the Senate basically sets the rules. Assuming they go with the same rules as Clinton impeachment went under, there are two steps, either can end it:
1) House and President both lay out their cases summarizing things and the house documentation is reviewed. The Senate then can vote to move to the next phase or acquit. To map this to a trial, this is discovery to see if there's any basis for the charges.
2) If the vote on #1 is to move to a true trial phase, then off we go for weeks.
This is an interesting timing since ALL Senators have to be in their seats, which means those running for election are off the campaign trail for the duration. This basically gives Biden solo attention on the campaign.
