It's not the falling asleep, it's them randomly running around the bed in the middle of the night. Hopping on, off, on the headboard, trying to lay on you...
People say they sleep well with them but sleep studies have shown you probably get worse sleep with them in the bed.
Years ago, I slept on my side curled up like I've just witnessed a horrible murder (still do), and the person I shared a bed with was only 5'4/163cm, and slept with their legs elevated, making them effectively shorter. Three dogs and a cat also shared the bed with us, an American queen sized bed.
We turned the bed sideways, used mattress toppers on each of our sides, and that created a spot in the (now much wider!) middle that was shorter than the rest of the bed. Affectionately named the Bed Trench, it was the new home of the pets.