Highly variable. If they departed on less than stellar terms i.e. contract not renewed or there was friction between other colleagues, then I would not expect them to be public about their next move. If however the circumstances were more genial e.g. spouse/family reasons or the job was not in that person's ideal geographic location and they made a long distance move just to get the job then it would be more likely to expect them to be public about it.
Ultimately, I think it comes down to the individual i.e. Did they have a more open/gregarious personality and make friends in the group? Or, were they the type to just read their slides and scuttle home? In any case, I really don't think there is optimal vs suboptimal way to tactfully handle whether one chooses to inform people of their future position. If they move on, they move on. In my opinion if they choose to not publicize this information I wouldn't say it's showing poor form necessarily. Likewise, if they choose to share it I would appreciate them sharing it with me, but that's about it.