What KPI's do you share with core product managers? How do you divide responsibilities for these KPI's?

This depends on the stage of the product. If the product has a solid product-market fit, the KPIs would be around User growth, Engagement, Retention & Revenue. Its always a best practice to have shared KPIs across both roles. 

With respect to responsibilities - a core product manager will continue to own the functionality of the product (capabilities, reliability etc). The growth product manager is always thinking of driving awareness, usage and engagement on the product.

There is usually one Northstar metric for the entire company, but each team comes up with its own KPI that feeds into the Northstar metric. For example, in a growth process where you have user acquisition, engagement, retention, and referrals, each team might be responsible a piece of this process that all end up contributing to growth. For example with Airbnb, their Northstar is "nights booked". However, they have one team with a KPI to grow user visitations, and another team's KPI is growing the number of listings. Their KPIs might be different but they both contribute to the Northstar metric of Nightly Bookings.