With Sienna, it's been kind of a process, not an all-at-once lightbulb moment. When I first got her, she had no toy drive at all. After a lot of work on retrieving, and giving her Kongs, etc., she did a few things:
1. She would take a toy if I handed it to her, then take it to her "stash" - she hoards everything in her bed. During the day sometimes, I would leave the door to the backyard open and she would take her stash, one by one, out to the yard, put everything in a pile, and lay in the middle of it. She never actually chews on the chewies, but seems to like owning them.
2. Maybe 10% of the time, if I threw a toy, she would run in it's general direction, then do zoomies around the yard. Gradually, her "chase" ended up closer to the thrown toy and now (2 years later) she will pounce on it 90% of the time. Whether it ends up in her mouth or not depends on a lot of factors, but it's about 50%.
One day she brought me a kong out of the blue. I was floored. It doesn't happen often, but she will now sometimes bring me a toy when she's excited, like when I come home or when we finish a training session. After practicing going to a mark, for instance, a few times when we're done she'll run over, grab a kong, and put the kong on the mark. I have no idea what's going on in that furry little head sometimes.
So to answer your question, most of the time I have to initiate toy play, but on rare occasions, Sienna will do it, and she had zero toy drive in the beginning. I think it's possible. What worked with her was rewarding any toy interaction at all. And I've been working on toys, fetching, etc. for literally 4 years. Only just this year have I seen her start to get really, really excited about it.
