No. PSP "game sharing" (using one PSN account on multiple PSPs) stores downloaded games on the PSP's internal flash memory (NAND) or a Memory Stick—not required for the sharing mechanism itself.
How PSP Game Sharing Works
Sign into the same PSN account on up to 2 activated portable systems (PSP / PS Vita / PS TV) simultaneously (Sony's current limit).
On each device: PlayStation Store → Download List → Download the game.
Game installs to Memory Stick (if inserted) or System Storage (PSPgo / internal 16 GB on 3000/Go).
Play on either device—only one can play the same title online at a time; offline both can.
Why a Memory Stick Is Still Recommended
Capacity: PSP-1000/2000/3000 have 0 MB internal game storage (only save data). PSPgo has 16 GB internal. A Memory Stick Pro Duo (up to 32 GB official, 128+ GB via microSD adapter) is essential for any library.
Speed: Faster load times vs. UMD.
Portability: Move saves, homebrew, ISOs between PSPs.
Caveats
PSN Store on PSP closed July 2021 (web store remains for purchase; download list still works on device).