The first thing that comes to my mind is radio. A radio could be pretty low-profile. You'd run the antenna, made out of some kind of organically-growable conductor like melanin (which humans already produce for other purposes) or iron crystals, etc., down the spine, parallel to the spinal cord, and power it with a thin layer of electric organs also running down the spine (similar to those of an electric eel, for example).
If you want it to require physical contact, though, then the best option is probably something that allows direct chemical communication between nerves from each person's nervous system, using neurotransmitters just like communication between nerves in the same body. You're going to want a lot of surface area so that you can match up a large number of parallel nerve endings. It's probably completely impractical to expect that you can accurately line up a sufficiently large number of nerve endings between two people exactly the same way every time, so there's going to need to be a good deal of processing that goes on an initiating contact to figure out what kind of information is coming in on which nerves, and you'll want to minimize the burden of adjustment and renegotiation, so you'll also want something that has a lot of friction to resist nerve endings sliding out of place once the interface is engaged. Fortunately, those two constraints complement each other nicely.
The interface will also need to be damp, so that there is an aqueous environment for neurotransmitters to diffuse through. And it would be very sensitive and easy to damage, so it'll need to be kept carefully protected when not in use. Between those two constraints, that means you're really not going to want to, say, build it into the hands or fingers.
There is limited precedent for this kind of thing in chemical communication between plant roots, or mycorrhizal networks.
The obvious solution is to use modified sexual organs! There's already a ton of nerves going to those areas, and an aqueous environment, and we're already pretty good at keeping them protected when not in use. They'd just need some physical modifications to ensure that nothing moves once engaged- a problem which many organisms have solved for completely unrelated reasons.
Of course, that approach has some major downsides, as well, like only being able to interface between people of complementary sex.
The best solution is probably something remarkably like the tentacle organs of the Na'vi, from the film Avatar. You've got a literal biological cable coming out the back of the back of the head or the neck, connecting directly to the brain. At the end, there's a retractable sheath which protects a bundle of tendrils, each of which has thousands of smaller cilia, which can intertwine with another person's tendrils to form a high-surface area, high-friction interface which is nevertheless easy to disengage when you want to, and there are no distinct "male" or "female" components so there's only one set of equipment that growing and protecting and anybody can interface with anybody else.