I recently did a walkthrough with the electrician who is helping me get my basement wired up. During that time I believe he had told me to pick up approximately 10ft of 4/4/2 electrical wire to run a 100A Breaker from my 200A panel to a 100A sub panel. Another guy told me I should get 4/3 wire because there shouldn't be a need for a ground on the sub panel as it would just create a loop. Both guys are great with electrical and from my understanding, know their stuff.

What wire size will I need to run from my 200A main breaker to the 100A sub panel?

Create a loop??

The guy who told you to get a wire without a ground does NOT know his stuff.

I assume because of the short 10' distance this is within the same structure as the main panel.

Previously any sub panel within the same structure as the main panel required a "4-wire" feeder. Two hots, neural and a ground. Detached structure sub-panels had exceptions to this rule.
In the 2008 NEC ALL sub-panels require a 4-wire feeder.

Ground is usually assumed in electrical work.
Just as 14/2 is 2 conductors is #14 with ground an this usually NM-B cable which implies the ground.

Thus in your notation, the cable type is not specified. We can assume that it's NM-B.

Thus 4-4-2 allows a higher neutral current than 4/3. 4/3 is really 4-4-4 in the original notation. There is never any 2 conductor cable with two different wire sizes.

4-4-2 is better than 4-4-4 and both likely contain a ground.

I get 1-1-1 with a minimum of #6 ground

Subpanels need the ground. Ground an neutral must not be connected in the sub-panel. Additional hardware may be required as well as removing a ground bond screw and purchasing an additional ground bar commonly called a "ground bar kit".