Won't you possibly get some information about some passengers by doing this, if you know / can figure out the distribution of the random numbers?

E.g., say that the random numbers are uniformly distributed between -200 and 200. If somebody says a number like 425, then I know they weigh at least 225 pounds. And the probability they weigh more than 225 - k pounds is 1 - k/400.

Yes absolutely, but you can just make the range wide enough to wash out the difference (like -2000 to 2000, summing to zero)

Then you would probably get more information from looking at the person thank you could from hearing a weight number that could be +/-2000 pounds

You could also do it modulo some number, I think?

Then knowing any individual number (weight + pad) mod n gives zero knowledge (I think. If all but one party reveal their pads, they can trivially determine the other's weight so it fails. Believe it to be secure if only 1 party is malicious, though. Cases for other numbers of bad actors are more complicated...)

Could use a similar technique remotely using a random number generator. Provided there are enough passengers the sum would approximately equal the true total weight.

There might be behavioural issues here though - perhaps people would keep generating a random number until they are provided one that makes their weight look 'better' (and 'better' probably means 'smaller' for most people).

Edit: comicjk made a great point about large numbers. Perhaps using large random numbers (eg a normal distribution with a mean of 4550 and a std deviation of 20-30) would help reduce the behavioural issue.

I like this better than my solution. The technique of adding random numbers has a more crypto feel to it. It would help to use big numbers, so that they would conceal the underlying number better (clearly single-digit numbers would be no help, might as well extend that and use 5-digit numbers).

This doesn't actually hide the information in edge cases.