Open source the software (in my opinion, you should release all of the code under a GPL compatible license (or public domain)), but not the contents of the databases that include private data (although the database schemas may be made available, and, depending what you are doing, maybe also the contents of the sqlite_stat1 table (or the equivalent for other database systems) if you are willing to include that information). Databases containing public data should be made available though, if you have such thing.
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Open source the software (in my opinion, you should release all of the code under a GPL compatible license (or public domain)), but not the contents of the databases that include private data (although the database schemas may be made available, and, depending what you are doing, maybe also the contents of the sqlite_stat1 table (or the equivalent for other database systems) if you are willing to include that information). Databases containing public data should be made available though, if you have such thing.