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). [As far as I can tell](https://en.wikipedia.org/wiki/Category:NATO_membership_referendums), referendums for joining NATO were held in a few countries as they were applying to join, but not in any of the countries that were already members of the alliance at the time. I might say that was unlike EU expansion referendums in some countries, [e.g.](https://commonslibrary.parliament.uk/research-briefings/cbp-7570/)

> France held a referendum on the admission of the UK, Denmark, Ireland and Norway to the EEC in 1972. An amendment to the French Constitution was introduced in 2005 which required a referendum to be held on any new EU accession, but in April 2008 the French Government approved the removal of the 2005 obligatory referendum clause.
>
> The Assembly voted in May 2008 to approve an amendment making it compulsory for France to hold a referendum on large countries joining the EU. This was seen as targeting Turkey in particular. In June 2008, the Senate voted to remove the constitutional requirement for a referendum for new accessions. In July it approved a bill allowing the President or Parliament to decide on a nationwide referendum.

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> Wouldn't the prospect of future wars be significant enough of a topic to require public consultation?

Well, decisions to enter "wars" (or better said armed conflicts, since wars are seldom officially declared theses) are typically not done by referendum either. Of course, ["public consultation"](https://en.wikipedia.org/wiki/Public_consultation) is a somewhat broader notion. At least in [the EU](https://medium.com/ecajournal/public-consultations-an-essential-tool-for-bringing-the-eu-closer-to-its-citizens-e33430dce747) and [the UK](https://publications.parliament.uk/pa/ld201719/ldselect/ldconst/27/2706.htm) there's a somewhat formalized notion of public consultation before legislation is adopted, even if that stops short of referendums on most issues.