The method below is the best way to reconstruct your library. If done correctly it will reproduce all your folders, labels, and even starred papers in your new library.

This method will not restore your current shared folders, and some duplicate references may be lost, even where they have previously been flagged as not duplicates.

TL;DR

The Guide below is fairly extensive, so here is the short version for when you are not worried about reproducing your folders/label and getting every detail right:

Preparation

Create a new account and Chrome profile

In general, paperpile accounts should be created under a personal Gmail account. Institutional accounts tend to come with an expiration date which forces a migration. Given the work involved in migrations, it is best to keep your research in an account you can maintain control over indefinitely. You can still use site licenses on your personal account if your institution subscribes.

Once you have chosen the email account to migrate to, create a new Chrome profile. This can be done by clicking on the name displayed in the upper right of the Chrome window, selecting "Manage People" and then clicking "Add Profile". The new profile will have its own Chrome window with no knowledge or shared resources with your other Chrome profile(s). Creating this separate profile is important to ensure that the syncing process goes smoothly.

Clean up duplicates

This method of migration does not import duplicates. If you have duplicates in your library, it is therefore important to first merge them so that you can better trust the results of the migration. To do this go to All Papers, then click the Duplicates filter in the right-hand panel.

Merge the ones which are actual duplicates by selecting them and clicking "Merge Selected", located above your list of papers.

Do NOT click the "not a duplicate" link for the others. This migration process does not respect these markers and so some of your marked references may not be imported. Instead, edit the references until they are no longer marked as duplicates, this includes at a minimum ensuring they do not have the same URL or identifiers (doi, pmid, etc.), but may involve more extensive changes to the title, author, and/or year.

If you know you have already marked many papers as such, there is little direct recourse and your best option is to compare the counts of papers in the different folders to ensure that they match.