Taking the Work programme (WP) wording as reference: “Implementing partners means third parties receiving financial support from the beneficiary and implementing the MSCA COFUND Doctoral or Postdoctoral programmes” (p.75 of the Horizon Europe MSCA Work Programme). An implementing partner that employs researchers but does not get funding would not strictly fall within the WP definition of implementing partners.
However, if a partner plays an important role in the implementation of the project but does not receive financial support from the beneficiary, the situation will be assessed by the evaluators at the proposal stage. In this case, the partner could still be considered implementing partner and still, partners recruiting researchers and playing a major role in the implementation of the project would need to comply with the specific eligibility conditions (e.g., country eligible for funding). The corresponding EU contribution (researcher-months) should be properly used by the beneficiary in some cost of the project though.