The requirements don’t stop at the manufacturer, the new delegated regulation puts some onus and responsibility on the Importer too:

Ensure manufacturer has completed their obligations (e.g. labeling, CE Marking etc.)
Provide trading name, address, website etc on product, packaging or accompanying documentation
Ensure while under their responsibility it is stored and transported such that it doesn’t jeopardise compliance
Where appropriate (with regards to the risks of the product) carry out sample testing
Keep copies of Declaration of Conformity for 10 years
Take corrective actions when known to not be compliant and respond to audits

Distributor obligations are largely the same (except the need for adding trading name and details), however, if a product is placed on the market under the trading name of a distributor then they take on the responsibility of fulfilling the manufacturer obligations.

Small Print: These posts are a summary only, include my personal interpretation and are not intended to be a replacement for reading the actual rules.