Since "Wand of Shocking Grasp" is not an item in the DMG or any other supplement I'm aware of, you'll have to ask your DM how it works; we can only speculate. (You should learn what you need to know about how the item works when you use identify on it.)
In general, items that specifically cast a spell but can be used by non-casters, which includes most of the wands, will specify a fixed attack roll or saving throw DC (as appropriate for the spell in question). As an example:
Circlet of Blasting
While wearing this circlet, you can use an action to cast the scorching ray spell with it. When you make the spell's attacks, you do so with an attack bonus of +5. [...]
There is no question as to what the roll should be: It's stated right in the item description.
If the item uses the wielder's spellcasting modifiers, as in the cast of most magic staff items, it will require the item to be attuned to a spellcaster or be used by a spellcaster, so the question is moot.
In the case of a character who can use the item despite not being a spellcaster, there's a specific rule found in the Activating an Item section (DMG p.114):
Some magic items allow the user to cast a spell from the item. [...] A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item. [...] If you don't have a spellcasting ability - perhaps you're a rogue with the Use Magic Device feature - your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.
A wand is not a weapon and does not use the Attack action, so it's not going to qualify for two-weapon fighting or any other attack-related abilities.