Can we call a person who loses things a "loser"?

The word to use for people who lose things would depend on why they are suspected of losing them. They might be absent-minded or forgetful as suggested in another answer, but they could also be disorganized, distracted, or preoccupied, for example.

Loser is not valid because here "lose" is always understood in the sense of winning and losing a competition. It can be neutral, to describe the loser of an organized game or prize, or to diminish those not at the top in business or society, or disparagingly to call someone hapless or a failure.

The only time it does not is when making a play on words. For example, the television show The Biggest Loser features teams of obese people who compete to lose the most weight. The biggest "loser" is thus actually the winner of the competition.

There are numerous other examples in which the obvious noun formed from a verb has a different and stronger primary meaning:

waiter > one who waits, as for a late train

waiter > one who waits on tables, as in a restaurant

boxer > one who packages things in boxes

boxer > one who fist-fights for sport

skipper > one who skips instead of walks

skipper > the chief officer of a ship

No, you can't, unless you want to be misunderstood. It will never be taken to mean someone who loses things. You just have to say such a person is forgetful, or absent-minded, or . . . always loses things.

It isn't common, but you can, as long as you specify what they lose: a loser of keys, the loser of those sunglasses, etc.