I was shocked when I saw umpires making decisions for Run out/Stumped without a third umpire in classic matches. So how do umpires make these decisions without a third umpire?
The same way they did before the existence of the third umpire, the same way that the vast majority of cricket is played. The officials learn the laws and best practice for umpiring. Then they pay attention to the game at hand and make observation of the events, determine a course of action based on combination of all the previous points, and effect a decision.
The umpire makes the decision. Whatever the umpire decides whether it is wrong or right has to be accepted, so basically the umpire's ruling is final. The umpire is trained to spot run outs so MOST of the decisions they make are correct.
Usually they to consult both the team players which are involve in that specific run-out. They ask both the team players and they discuss it if there is a voting tie or any doubt it goes with the batsman. Usually you hear a line from commentators that the doubt will be in the favor of batsman. For Six or Four they need to ask the audience.