I received a letter from my POI a year or two ago - with a requirement to confirm that I complied - specifically to clarify that another operator's training does not meet the requirements of an operator to provide training to its crew. The issue seems to be that the rules require that the operator shall provide training i.a.w. its approved syllabus. Another operator's training isn't part of the subject operator's approved syllabus. You don't have control over what is taught or not taught, including operator specific stuff.
If you contracted with the other party to provide training, then that would be different. You would still be responsible for the outsourced training though. No different to contracting with Simcom/CAE/FS to provide type specific training. What ever training they give is held to be your approved course, not theirs (even though they may have provided the syllabus, ultimately it's approved as your syllabus, not theirs).