>If private individuals aren't meant to interpret the Scriptures on their own, who is?<
The point Peter was making is that scripture is not a production of the human mind as you implied but is "God breathed". The Greek word translated interpretation is better translated origin in this passage when put into context with verses 19 and 21.
Num 16:28 And Moses said, Hereby ye shall know that the LORD hath sent me to do all these works; for [I have] not [done them] of mine own mind.