We know, that by adding be-, we can make intransitive verbs transitive. Can we conclude, that all be- verbs are transitive?
Depends. In general, you can assume that every word that you create using this word formation rule will result in a transitive verb. After all, that's what it is good for. However, you can never safely assume that every word that has the structure of a certain word formation rule will always have the function / meaning which is common for that rule. Some words may lose their original meaning over time, or be remnants from a time when the formation rule worked differently; however, I do not know if there is an example for be-.
Transitive verbs, in the definition used by Canoo.net, are those that take an accussative object, and thus those which, when transformed into the passive voice, retain a subject in the passive voice.
Begegnen fulfills neither condition. Jemandem begegnen uses a dative object as does etwas begegnen. A passive voice is only really possible for the second one:
Dem Vorwurf wird begegnet.
You see no subject in there, thus the verb is intransitive.
"Bedürfen" is not transitive since it takes a genitive object: "Ich bedarf deines Rates". No other option even in colloquial speech except for switching to some other verb like "benötigen".