No it doesn't appear so. If you look under the stat information on the right while cycling through it seems to have independent stats for both. Unless you're referring to things such as STR and Arcane in which case I think so. Crit Damage and % are independent to said weapon however.
I mean any bonus stat. It sounds like you are saying some are global and some aren't. The stat system is pretty confusing in this game since the stat screen only shows a small subset of your stats and the fact that you have 2 weapons that give bonuses with no indication of if they stack.
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Some definitely do, others don't seem to do it. Depends, really.
Achidna's Fangs for example, have Searing Venom as the special ability, which grants what is similar to "health on crit". Point is, it's not the same as that actual ability, since SV works on any weapon, as long as you have AF equipped. So that ability is shared across all weapons (yes, including the guns), not just the actual AF themselves. A normal "health on crit" on a weapon only works when using that weapon, it doesn't carry over to the others.
My guess is that standard abilities don't carry over, unless it's a character stat boost (+STR, +health, stuff like that), and legendary special abilities may do so, depending on the actual effect.