I'd say the other characters decide on the object, otherwise it would say "take one object from each character".
I think this is correct. Only cards like Sneakthief let you choose objects from other characters.
In Beggar King card other characters decide but I wonder if he can give for example Runesword to a Good character which he must immediately ditch?
Or he must give an object which other character can take?
I also think everyone interpreted the card correctly.
The implications of other players giving what they want gives way to many sinister outcomes for the happy 5-6 roller. You'll receive a bundle of useless items most of the time, but I don't think there's anything that forbids giving an Object that the receiver must ditch. Or even worse, giving a Living Doll, Lodestone, False Grail. I would hold onto my Runesword, though.
There are so many good cards in the City deck that a totally negative Beggar King is more than welcome in the balance.