Should I always equip a weapon in my free hand, even if it is a weak one?

Solution 1:

In short: you should be dual-wielding if you have an off-hand weapon with many powerful effects; otherwise use a shield.

When dual-wielding the effects from both weapons are added together and applied with each attack. So if your off-hand pistol has +X life-steal, then all of your attacks will have life-steal.

However, DPS is only the better of the two weapons (discovered with a little testing). Attack speed averages because attacks alternate between the two wielded weapons, each taking the time of the respective weapon. For example, if you have a quick pistol but a slow wand, the pistol attacks will happen very quickly after each wand attack, but wand attacks will happen slowly after each pistol attack.

What this means is that you should only dual-wield if you have an off-hand with a fast attack speed and many useful effects.

The Dual-Wield skill increases damage significantly, but it is geared towards Berserker as most of his skills add the DPS of both weapons to skill damage. Being an Alchemist, this skill should not interest you nor affect your decision.

Usually the defense of an off-hand shield is more valuable than a weapon that only adds its value in effects. Also, as your mana regeneration increases you will be using only skills, meaning your weapon is never used whereas a shield is (passively).