Do owners of reentering spacecraft notify the countries' whose airspace they are likely to violate and seek permission?
No, there is no requirement for notification, and in most instances there is no notification.
There are three main guiding principles of international space law that could apply.
The Outer Space Treaty has a vague reference that could be relevant:
States Parties to the Treaty conducting activities in outer space, including the moon and other celestial bodies, agree to inform the Secretary-General of the United Nations as well as the public and the international scientific community, to the greatest extent feasible and practicable, of the nature, conduct, locations and results of such activities.
The Registration Convention only covers launch with no treatment given to reentry.
The Liability Convention only covers claims after damage.
Local authorities like the FAA and FCC may impose license-based restrictions on reentry due to their own policies. For instance, in the U.S. there are guidelines that objects that have a high casualty probability must undergo a controlled re-entry.
In case of controlled re-entries, countries will normally issue a notification to overflown states, although, more often than not, the reentry will be over the ocean. Without a controlled reentry, which is the case for most small/medium satellites, notification is close to useless. The error bars on a natural reentry can be several orbits. In other words, you may need to notify the entire world that you are re-entering.
Having said that, Space-Track will issue "Tracking and Impact Prediction" (
TIP_msg) messages for any object whose reentry is imminent. First prediction can be issued up to 60 days before predicted reentry (
60day_msg) although often it is only a week or two ahead of prediction. They will then continue issuing updates on a daily or higher basis.
To illustrate, I pulled up the Decay Data for object 8418 (1975-081D), a booster stage launched by the Soviet Union in 1975. First
60day_msg was issued 2019-02-28 with a predicted decay of 2019-03-16 [You can see 60 day is a misnomer]. There were 7 subsequent
TIP_msg warning of imminent reentry. Actual reentry occurred 2019-03-24, as captured by the
decay_msg. I'm attaching a portion of the last
TIP_msg below. It gives you an idea of how the predicted decay epoch varied from 2019-03-19 through the end on 2019-03-24. Decay orbit varied between 31402 and 31409, ultimately most likely coming in on 31407. The final
decay_msg does NOT state the actual decay orbit, just the day of decay.
Moral of the story, practically impossible to know exactly when it is coming in, and by extension who to notify.
Object Message Epoch Decay Epoch Rev Direction
8418 2019-03-24 04:01:00 2019-03-24 02:39:00 31407 ascending
8418 2019-03-24 00:37:00 2019-03-24 02:42:00 31407 ascending
8418 2019-03-23 21:59:00 2019-03-24 03:54:00 31409 descending
8418 2019-03-23 02:54:00 2019-03-24 01:20:00 31408 ascending
8418 2019-03-22 00:49:00 2019-03-24 00:36:00 31406 ascending
8418 2019-03-20 19:44:00 2019-03-23 20:09:00 31404 descending
8418 2019-03-19 23:16:00 2019-03-23 16:51:00 31402 ascending