Yes, some sure do. And others also do it unconsciously. In other words, they engage in 'grooming' type behavior but it's not with any apparent intent to take the next step. They wouldn't even necessarily think of it as grooming.
You probably don't need me to tell you that there are all kinds of variations. Heck, one of my more interesting case studies got his biggest kicks from merely securing a person's name, phone number, and address. Until he managed to get that his stalking could be fairly intense, but once he was confident he had the correct information he pretty much filed them away and all but lost interest and moved on to the next target. In his mind, they were now his 'possession' in a sense and that was sufficient.
The fat unused "black book" gone wild. Whatever.