No, you actually don't have to do the testing, except in some special circumstances (e.g. childrens' toys). Generally, the "CE" mark is a self-certification done by the manufacturer or importer that certifies that the product meets certain (category-specific) criteria. You can do the proper thing, test your product and then certify it. But it is also possible and usually legal to just issue the certificate based on a self-assessment and deal with the fallout later, if there ever is any. That's how basically all non-EU manufacturers and importers get their CE mark...
https://europa.eu/youreurope/business/product-rules-complian...