I often hear people referred to as "der Paul" or "die Barbara". So I wonder whether it is correct to say "Das Buch von Paul" or "Das Buch vom Paul"
Note there are two levels of colloquiality/regionalism in your examples:
The lighter colloquialism is the replacement of the proper genitive form
by a prepositional construct using von and the dative:
das Buch von Paul
which is somewhat accepted in standard German, but especially in dialects that do this all the time.
The second level of colloquialism is the usage of a definite article in front of a proper name. This is in no way standard German, but still common in a lot of dialects:
das Buch von dem Paul
das Buch vom Paul
The bandwidth of "how correct" both constructs/the combined constructs are considered to be varies from "colloquialism" over "regionalism" to "dialect", depending on what book you read.
Absolutely "correct" (by the book) is only