You don't need to get separate paper tickets to transfer across companies, but you do need to get a special transfer ticket, which can be complicated. The good news is there's really no reason to deal with them these days, just get a Suica/Pasmo card and tap on/off.
The one complication is that there are extensive through services, meaning companies running on each other's tracks. This is super convenient, because it avoids having to transfer, but can be confusing. You can board a train in the Yokohama Minatomirai line and travel smoothly through the Tokyu Toyoko, Metro Fukutoshin and Tōbu Tōjō Lines to emerge right on the other side of the metropolis.