Are we talking specifically about the law? I would never use "possessor" in speaking or writing, but I think it is used in the law.

"Proprietor", in normal speech, is used only for someone who owns a business, usually a hotel or a bar (US usage).

"Owner" is used for everything else (and it can also be used for owning a hotel or bar).

The concept of a "Besitzer" who is not an "Eigentümer" is a bit difficult to express in English. You can say, for example, "It's not my book--I borrowed it from John." I suppose that makes you the "possessor" of the book, but you would never say it that way.

Thanks indeed GeorgeA. So would you say that "owner" and "proprietor" are used for "Eigentümer", but not "Besitzer"? And if you want to point out that you refer to a "Besitzer", would you then use a more specific word such as "operator" for someone operating a business without being the owner at the same time?

Your question made me wonder if I was wrong about the English meaning of "proprietor". Here's what Merriam-Webster has to say:

1: one granted ownership of a colony (as one of the original American colonies) and full prerogatives of establishing a government and distributing land
2 a: a person who has the legal right or exclusive title to something : owner
2 b: one having an interest (as control or present use) less than absolute and exclusive right

So, according to 2a, "proprietor" means "owner; but according to 2b, a "proprietor" many just have an "interest". Confusing.

The person running a small business could be called a "manager"; for a larger business (especially a corporation), it might be "Chief Executive Operator" or "Chief Operating Officer".

Be|sitz, der; -es [ 15.J. ü . ]: a) (materielle) Güter, die jmd. geerbt od. erworben hat, sodass er darüber verfügen kann (im allgemeinen Sprachgebrauch meist gleichbed. mit Eigentum gebraucht, juristisch davon unterschieden als das, worüber jmd. die tatsächliche (nicht unbedingt aber die rechtliche) Herrschaft hat)
© 2000 Dudenverlag

So Besitz is technically possession, not ownership, in a legal sense:

Longman Dictionary of Business English:
possession (law)
the state of having the right to use and enjoy a thing such as a house, whether by right of ownership or not
ownership (law)
the right to hold a thing entirely as one's own, including complete and permanent control over it
proprietor (law)
1. the owner of land and buildings
2. the owner of a business, esp. of a hotel, restaurant or newspaper
3. the owner of a copyright, patent or other legal right such as a brand name or trade.mark

Thanks to everyone for your explanations, that was of great help!

Would you agree that it is not quite right to translate "owner" or "proprietor" with "Besitzer" as LEO (amongst others) suggests?

#5 - Yes, but only in a technical, legal sense. As Duden says, people don't usually make such a precise difference, and "possessor" is not very common at all in normal speech, where "owner" is more appropriate IMHO.

(Even in legal texts it might be worth changing the noun into a verb if that's possible, to avoid that "possessor", or using something like "the person/party in possession of the whatever". But I'm not a legal expert, should we call in Bacon?)

New SOED:
possessor
A person who possesses something; a person who holds something as property or in actual control; a holder; an owner; Law a person who takes, occupies, or holds something without necessarily having ownership or as distinguished from the owner.

This could do with some marking in Leo, though - for "owner" etc there is no comment like the one when you look up "Besitz":
possession der Besitz - Sachherrschaft, aber nicht Eigentum
Dictionary: besitz

Thanks again CM2DD for your efforts. No need to trouble Bacon I think - unless he/she has time and interest. :-)

"Yes, but only in a technical, legal sense. As Duden says, people don't usually make such a precise difference, and "possessor" is not very common at all in normal speech, where "owner" is more appropriate IMHO."

Yes, that's what it looked like to me as well, but I was not sure. Thanks for confirming!

"This could do with some marking in Leo, though - for "owner" etc there is no comment like the one when you look up "Besitz":
possession der Besitz - Sachherrschaft, aber nicht Eigentum."

That wouldn't be bad to make it clearer, would it?