On top of the answer by Matt, it seems to me there is one additional reason: Money laundering.
It's an open secret that a lot of these arms trade use very opaque pricing and payment scheme. Lots of room for paying off cronies (both abroad and local) and 'renting' influence.
Even in very dysfunctional countries like those in the gulf the ruling tribe does not rule alone. The Power structure typically involves various groups with competing second order interests. The ruling clique can often not simply siphon money out of the state coffers to buy off/reward support from one group too openly because of the risk of antagonizing the other competing groups in the power pyramid.
A certain level of opacity also keeps everyone guessing about the actual reward to loyalty, puts the ruling Prince in better control of the bribing rate and gives him plausible (relatively speaking) denial-ability.
A lot of these arms firms are also large multinationals with intermediaries in various countries (including banking safe heaven) with legitimate and large cash transfer between them. The ruling clique in effects rents out these infrastructure to launder money in a way that would be much less traceable than a simple bank transfer from state coffers to a swiss bank account. This reduces the risk of having the money clawed back come the next coup/round of economic sanctions, a very valuable advantage indeed.
It's a bit dated now, but you can find examples of how this was done in a more naive era in the more publicly disclosed aspects of the infamous Al-Yamamah deal.
This explanation also accounts for other non-arms opaque purchases favored in the region (big construction projects, big ticket items with opaque pricing like civilian airliners, ect...). A predatory state has to do what a predatory state has to do, I guess.
These deals are typically large and involve western politicians and protected/connected industries (like the defense industry) from the onset. This shields them from public inquiry in the west.
Of course, it's a two way street and securing these deals require some impressive feats of cultural flexibility from western politicians, captains of industry and bureaucracies (from the wikipedia above):
Also, the role of Sir Dick Evans, the BAE chairman at the time, in
winning the Al-Yamamah arms deal should not be underestimated. It is
believed by some that he managed to secure the deal by his ability to
swallow sheep's eyeballs as though they were cocktail canapes at
banquets, thus impressing his Middle East clients.