The person is employed by one organisation and does their work for another. Both sides can get the relationship wrong, in opposite directions, and both errors are common.
Claiming too little
Treating the person as a resource rather than as a member of the team. Not inviting them to the team meeting. Not telling them what the company is doing. Never explaining what happened to the work they produced.
This is the commoner error by a wide margin and it is the one that shows up in the attrition figures. Somebody who has processed four hundred invoices without ever learning whether the month closed cleanly has been given no reason to care whether it did.
Claiming too much
Acting as their employer. Promising a promotion, a bonus, or a job. Conducting a disciplinary conversation. Asking them to do something their employer's policies prohibit. Contacting them outside their working hours as though the arrangement entitled you to.
The entry on feedback sets the line: talk to the person about the work, talk to the provider about anything touching employment.
The single most valuable thing
Tell them what happened to their work.
The report went to the board and here is what was decided. The campaign produced this many enquiries. The reconciliation was clean for the first time in four months. It takes a sentence, most outsourced staff never receive it, and it is the difference between a task and a job.
Ordinary inclusion
The team channel. The team meeting, at an hour they can attend. The company update. Whatever you do for birthdays or for good news, done for them as well.
None of this is a gesture. It is the ordinary treatment your other staff receive without anybody calling it engagement.
Where the two loyalties genuinely conflict
Occasionally they do. You want overtime the provider's policy restricts. You want the person on a public holiday. You want them to handle something outside the agreed scope.
Route these through the provider rather than asking the person, who cannot say no to a client comfortably and should not be put in the position of trying.
A person who agrees to something they should have declined has done you a short-term favour and created a problem for their employer, and you will hear about it in a less convenient form later.
The one-person account
Hardest case. Somebody working alone on your account, in an office full of people working on others, belongs nowhere in particular.
The remedies are the ordinary ones applied deliberately: more contact than you think necessary, a genuine relationship with the on-site supervisor, and where possible another person on the account even part-time. Isolation is the most reliable predictor of a short tenure in these arrangements.
Meeting in person
Changes the relationship measurably and durably, which the entry on visits is about.
What the provider owes here
The employment relationship, the pastoral support, the career conversation and the social side of an office. Ask what actually exists rather than assuming, because it differs a great deal between providers and it determines how much of the belonging problem lands on you.
The test
Would you do this for somebody sitting in your own office? If yes and you are not doing it, do it. If no and you are doing it, stop.
Almost every question in this entry resolves that way.