The basic answer to this is : No.
Clearly, by the rules, you cannot envenom a weapon & give it to another person to use as an envenomed weapon.
Similar to the rule for envenoming arrows, if you lose track of a weapon that has been envenomed
(or lend it to someone to use <while> it is envenomed), you must assume that the poison has been
rubbed off or otherwise handled in a non-protected manner. The poison is gone.
By the wording of the skill, you may only use envenom effects that you place yourself.
Envenom effects cannot be passed off to another person, skilled or not.
This skill is mostly a offshoot of the KISS outlook for rules.
We don't require any phys-rep of the actual poison on the weapon - not even a tag / card.
Therefore, only the person with the skill may use it. Simply avoids confusion.
Keeps things simple.
-Chris