Generally, some combination of the acquired host or the buyer will communicate the details of the transition to the customers, outlining the upcoming change. A failure to communicate the change at all would probably be the first sign of a rough transition to come. Some post a notice on their front Web page, others will release the news in the form of a press release, blog post, email or forum thread. Whatever the method, hosting providers seem to understand that their customers need to know what's going on, especially when there's a chance of downtime or other minor service issues. Though downtime can be problematic, scheduled downtime can, at least, be planned around, and your customers can be warned.
The problem most likely to cause upset during the transition is downtime. Although it's possible your service won't go down, if your hosting provider has properly prepared, sometimes you'll encounter an outage. In most cases these problems will be fixed quickly, and compensation issued. In rare cases, downtime and instability will plague the transition. It might not be a bad idea to do some research on a suitable backup host in case this sort of problem occurs.
Be prepared for the possibility that other issues could crop up with your new provider delivering a different style of customer service, glitches in email servers or lost emails, billing system problems, or in rare cases, information being lost during the migration process.
Overall, things go smoother if your hosting provider is open and communicative during the transition. It's likely that they'll try to fix your issues as quickly as possible as soon as they're done with the migration process. Not all Web hosts are that transparent, however, and if yours isn't, decide whether you want to wait for problems to occur, or take some precautions in advance.