1) Track data while at work. Display only at work. 2) Track data while at work. Display at home and work. 3) Track data at home and work. Display at work 4) Don't track data at all.
Of course there's a few cases I missed but from what you said I don't see the reason they'd track me at home and display at work or any other various combinations excluded.
If I understand your question correctly, the answer would be 1. In most cases, they're doing cookie based tracking and you wouldn't have any crossover unless it was site specific retargetting and you've logged into an account on both platforms. If there's any crossover, its because you have the same browsing behaviors on both platforms.
Interesting, thank you for the information.
(Yes, yes, people shouldn't use UA sniffing to do that... but they do.)