Just so readers understand this clearly, the DMVs do not access private data (e.g. customer details), instead they access metadata (e.g. what queries take the longest time to run). So stating the data is private, is not a valid reason to preclude the use of DMVs on a production database.
If you have access to the production server, and if you know the troublesome query, its execution plan (typically) already exists, and can be obtained easily via the DMVs, there is no reason to run it again! If the query is running slowly, do you really want to run it again (slowly).
It is possible to find the state of the statistics (number of rows changed since last stats update, date last updated etc) via a simple query of some system tables, these are not DMVs but can be queried in a similarly (typically) innocuous manner. (It might be argued that the stats column values are ‘reflections’ of private data)
The above suggests there is no reason to leave the production database...