Can a private pilot fly commercial-type aircraft?
Your certificate level determines the type of operations you can conduct*, not the type of airplanes you can fly.  The types of airplanes you can fly are determined by your category, class and type ratings.

If you have a Private Pilot certificate with an instrument rating (not sure if this is required, but you'll want it either way) and an airplane multi-engine land category and class and a B777 type rating, then yes, that person can operate the aircraft privately.  

He will also need an SIC and he will only be able to collect the pro-rata share of operating expenses from people he carries.  He will also want to be careful in who he lets on his plane to avoid operations that are regulated under 14 CFR 119. 

An example of such a pilot is John Travolta, who has a private pilot certificate with the following ratings:

PRIVATE PILOT
  AIRPLANE SINGLE ENGINE LAND
  AIRPLANE MULTIENGINE LAND
  INSTRUMENT AIRPLANE

Type Ratings:
P/B-707 P/B-720 P/CE-500    P/CL-600    P/EA-500S
P/G-1159    P/HS-125    P/LR-JET

Limits:
ENGLISH PROFICIENT.
AUTHORIZED EXPERIMENTAL AIRCRAFT: SO-G2.
B-707 SIC PRIVILEGES ONLY.
CE-500 (VFR ONLY).




* Private pilots may only collect pro-rata operating expenses, Commercial pilots may be paid to act as PIC and ATPs may act as PIC in 14 CFR 121 operations.