This is the list of anything you can do and can't do in splitscreen coop especially Player 2 since Player 2 is more restricted then the main player (this is based on my experience, will add more if found anything new, or correct wrong information) :

•  Player 2 can't use any other quick items other than blood vile, the quick item menu is disabled for them, maybe this is intended so they can't waste resources however I want to share it, utilizing it to our advantage. (it's not intended, it's an unfortunate limitation due to how the code base was set up) Since Player 2 cannot use any other quick items, ultimately they can't also use Magic/Tools.
• Player 2 can't enter level transitions. (doors/gates/darkness that goes to another location)
• Player 2 can't interact with lanterns.
• Player 2 can't interact with the doll, the insight doors or the key-locked doors. (you can interact with locked doors but cannot use a key)
• If Player 2 dies,  it will return to the normal screen  until the main player returns to Hunter's Dream to respawn Player 2.
• If Player 1 dies, both of you will automatically return to Hunter's Dream to respawn, regardless if Player 2 is alive or not.
• Both player do not share bullets nor blood viles.
• If Player 2 respawns, they will be given the same amount of blood viles and quicksilver bullets as the Player 1.
• Game doesn't save Player 2's current attires, and goes into default attires.

NOTE THAT SHOULD BE THERE FROM THE START: THIS IS REFERRING TO PURE LOCAL SPLITSCREEN AND NOT VIA PARSEC, SORRY ABOUT THAT.
- Check below for the image guide if you don't know what I'm talking about.
(If you have 2 controllers, just plug them both and load a game, coop enabled.)
(If these are generic controllers/PS controllers, use X360CE and set them up there, check the image guide below for reference, or DS4Windows for DS4 controllers.)

(If you want to play splitscreen coop with keyboard as Player 1, controller on Player 2, use X360CE, set Keyboard to Controller 1, remap it or not idk, then set your controller to Controller 2, then load a game, after that, disable Controller 1 (untick Enable 1 Mapped Device). If it doesn't work, invert it, set Keyboard to Controller 2 and the controller in Controller 1, load a game, disable Controller 2.)

(Not disabling Keyboard (Controller 1/2) may cause issues with the controls, especially if you remap it to the same controls as the default keyboard controls (check the photos), pressing the same button two times simultaneously.)

(This is the only easy solution for now until she adds a proper menu on what controller to use for both players.)