The most basic form of operating would have one copy the checksum file made from running:
md5sum Dir1/*
over to the directory you are wanting to test validity of the copying done. (If you do this while also copying the other files when your backing up or the likes, then you don't have to do it separately with:)
cp checksum Dir2/checksum
cd Dir2
Changing to the second directory can make commanding things simpler, and if your likely to have to deal with missing files it will help assure you have the proper path to the files working in the terminal (and its history of commands) which can, if nothing else, let one copy and paste them into the command lines later.
md5sum -c checksum
Gives the integrity of that copied.