I serve on a presbytery committee that is looking for ways to share work and resources with other nearby presbyteries and finding it very difficult to get anything going (for the uninitiated, a presbytery is a regional grouping of Presbyterian congregations).
It’s not going to work. It shouldn’t work.
It would be different if there were a person, if there was a group of people, that wanted to work in something together. That’s how real partnership works: between motivated people.
But the pursuit of partnership as a practical idea, say for the purpose of consolidating diminishing resources and cutting costs, won’t work. And that’s good, because when it does work it’s imposed on people who don’t really want it and can’t really own it.
What work do we want to do, and who do we want for our partners?
YoRocko! 
I am on a similar committee and it has been the most frustrating experience of my professional life.
In San Francisco presbytery, they coined the unfortunate term “horizontal partnering” to talk about pairing up the resources and needs of different congregations. Luckily, the phrase soon passed into oblivion. You took me back to another difference by which I’ve always been struck: between intentionally trying to create community and celebrating community where it is graciously found.
Been thinking a lot about this the past couple of years as we work on partnering with another congregation. I think you are correct that real relationships (partnerships) are between real people and therefore organizations struggle to make it work. I am a little less despairing of local congregations acting as “persons” in their own self interest in entering into partnerships. Is there not some kind of democratic process where a partnership is entered into rather than imposed?
Our EP keeps telling me that presbytery as council is primarily about connecting with sessions as councils. I get this as a polity abstraction, but I don’t buy it in practical terms.