About Outsourcing Delivery
What would happen in the situation where a business project team outsources the analysis, development and testing effort?
First things first, the requirements and scope should be formalised with a signed business contract between the business and the outsourced delivery team.
In the case of having an outsourced delivery team, working on-site or off-site, the business project team is unlikely to be the appropriate team to be running the Behaviour Specification writing session, because they are unlikely to have all the necessary resources, nor a holistic perspective of the solution.
A better approach is to have the outsourced delivery team own and organise workshops to collaborate with the business and other team members in order to perform the business analysis and Behaviour Specification writing sessions. With the outsourced delivery team producing the Behaviour Specifications, they should handover back to the business the completed Behaviour Specifications as part of their documentation and test assets. The Behaviour Specifications and software components also can be used as a metric to track the progress of the delivery team.
In any case, with the requirements and scope already formalised, there is no need to also formalise the Behaviour Specifications with a signed business contract. Please see What Are Behaviour Specifications? for more information on what a Behaviour Specification is, and importantly, is not.