As of Update 10.3, the toxin mixture in the barrel is affected by the Acrid's energy color, and the tiny explosion effect on needle impact. Note that the explosion effect is purely cosmetic.
As of Update 11.3, the Toxin effect's DoT scales with Acrid's base damage, not including elemental mods. To clarify further, onlyHornet Strike and Magnum Force directly affect the DoT. Damage per tick is 50% of this base damage (a rank 10 Hornet Strike will yield a base damage of 112, with the DoT dealing 56 Toxin damage per tick).
Oddly, despite its incredibly high base accuracy the Acrid's accuracy value is heavily degraded by Magnum Force well beyond what the mod's penalty would suggest, making the weapon highly inaccurate as a result.
To clarify on how the weapon proc works: the weapon will proc toxin on every single shot, unless another elemental status procs. So if you mod the weapon to deal only corrosive damage using an electricity mod, the weapon will proc toxin on every shot unless the corrosive procs. The proc chance listed on the weapon is actually for any other elemental damage that is created by the weapon.