For binary selection, keep the watch and tell them to make an indication (say something) in a few seconds. Note down whether the seconds were even or odd, then hand them the clock and swap roles. Keep the watch hidden until the moment you need to inspect it; record the time as soon as you can see the watch. Merge the two notepads to get the final stream of randoms.
For last digit of seconds, wait a few minutes between each query (making sure they can't see the clock during the rest period). "A few minutes" is imprecisely measured by you, who does not have access to a clock. They delay would foil any attempts at them keeping track of the time in their head. Bonus for distracting them to help with that. And instead of asking them for the last digit, have them flip the watch and then record the time that you [both] first saw.
For binary selection, keep the watch and tell them to make an indication (say something) in a few seconds. Note down whether the seconds were even or odd, then hand them the clock and swap roles. Keep the watch hidden until the moment you need to inspect it; record the time as soon as you can see the watch. Merge the two notepads to get the final stream of randoms.
For last digit of seconds, wait a few minutes between each query (making sure they can't see the clock during the rest period). "A few minutes" is imprecisely measured by you, who does not have access to a clock. They delay would foil any attempts at them keeping track of the time in their head. Bonus for distracting them to help with that. And instead of asking them for the last digit, have them flip the watch and then record the time that you [both] first saw.