NATO expansions function like signing & ratifying any other treaty. It's entirely up to member countries own political system how they handle that. 

>  it is ok for politicians to make the decisions relatively privately?

Treaties are usually ratified by parliaments. Your beef seems to be with [representative democracy](https://en.wikipedia.org/wiki/Representative_democracy). It's true that in some cases referendums are called, e.g. Brexit, but it's a matter of judgement how important a treaty is before going to that level. 

It's more likely a referendum would be called when the matter is contentious enough. NATO expansions [generally have garnered little internal opposition](https://politics.stackexchange.com/questions/73266/when-and-where-was-the-parliamentary-nadir-for-the-1999-2004-nato-expansions).