How do countries trade weaponry with each other?

Not really, states are not direct sellers when it comes to trading weapons. To put it simply. Whether state-owned or privately owned, the manufacturing of weapons is ultimately a company in the direct manufacturing, the country in need of weapons is to send orders to these companies, but whether or not to commit to this contract still depends on the agreement between the weapons company and the country where the weapons company is located (more state-specific organs) or from the state (more state-specific organs) permission, and of course, to comply with the appropriate laws (awkward) laws (awkward) Regulations, provisions of international law, resolutions of international organizations (mainly arms embargoes), without the permission of the above conditions weapons companies can not commit to a contract.

As for how the state procurement personnel how to contact the manufacturing weapons companies, both to participate in exhibitions can be contacted, but also the weapons company's country as an intermediary to contact and other ways. All that's left is payment and shipment (note that some countries won't allow merchant ships with weapons to anchor, it's a law of the sea issue - transit rights of passage, contiguous zone of the territorial sea and all that)