If the government already has an open data initiative, then it’s like @Stephen describes.
In most cases, the question of which license to use is already answered, because many catalogs use the same license for all datasets.
For other datasets, there’s often a step of validating that there are no third-party rights over the dataset, but my understanding is that any government has all the rights to their own budget data.
The budget is prepared by the legislature, but at least in Canada, there’s a Treasury Board Secretariat (on the government side) that has the rights to publish this data. In other countries, it’s possible that the budget is entirely under the jurisdiction of the legislature without any government department having the rights to publish this data. This can make data release more complicated, because in general legislatures are not very aware of open data.
In terms of legal implications of publishing unlicensed data, it’s often a legal gray area. In some countries, data is not subject to copyright, so you would be able to publish it. In other countries like Canada, there are some specific criteria a dataset must meet for it to be subject to copyright. Anyhow, I know many organizations who republish government data that is unlicensed, and I’ve never heard of anything dramatic happening - mainly because it would be reflect very poorly on a government for it to sue a website for publishing publicly available, public interest data. In some cases, this unlicensed republication leads to the government applying an explicit license on the data.