The drivers side of the engine cams hold still and move with little resistance . The passengers side generally move off the marks when the belt is removed as they are on the lobe of the cam in the marked position .Top intake cam should only be wound clockwise to reline up and bottom exhaust cam only wound anti clockwise to reline up as per the manual as this is an interference engine design.
I use a breaker bar and socket to hold the cam in place as I put the belt on then if the belt is kept taught while you run it over the top and bottom cam wheel as well they usually hold in place just with the belt and some slight tension from your hand on the belt as it is put on . Start the belt on the drivers side first then work across the top over the passengers side and down. The manual outlines this.
If there is no tension on the passengers side cam wheels that moves them off the timing marks sounds like something is wrong . Only the drivers side is off the lobes and stays exactly where you put it.
Do you have a manual it clearly has diagrams and fully explains it if you follow its hard to go wrong . Once the belt is on dont turn the engine till you count the teeth between timing marks as outlined in the manual to make sure you spot on and not a tooth out.