Two separate questions
Custody in New York has two parts. Legal custody concerns major decisions, such as education, medical care, and religion, while physical custody concerns where the child lives day to day. Parents can share one without sharing the other, and many arrangements combine joint legal custody with a primary residence and a parenting schedule. Asking for joint custody without being clear about which part you mean tends to confuse the discussion, both in negotiation and in court. Physical custody can also be shared, with the child spending substantial time in both homes, though that depends heavily on distance and the child's routine.
When shared decision-making holds up
New York law does not presume that joint custody fits every family. Courts have been reluctant to order it when parents are so hostile that joint decisions would become a new battleground, and they also weigh which parent supports the child's relationship with the other. What tends to matter is the track record: how you have communicated about school, doctors, and schedules, and whether the child has been kept out of adult disputes. Keep co-parenting messages, school and medical records, and anything showing how the two of you have handled decisions so far. Ask your lawyer before posting anything about the other parent or the case online.
Building a realistic proposal
Agreed arrangements are common, and a well-drafted one is specific about how decisions get made and what happens when parents disagree. Some agreements give one parent final say in a particular area while keeping consultation duties for both. When we start, we look at your current schedule, the distance between the households, and how communication actually works between you. From there we can talk about what kind of shared arrangement is realistic and how it would be presented if the matter does not settle.