Man, talk about a perfect example of what Tom Hanks said in Forrest Gump: Stupid is as stupid does. Thankfully have not heard anything like that in the US, and specifically in Washington state where we are located. I am making the logical assumption that since we are getting the first dose of the vaccine at a hospital, they have enough (maybe more than enough) for the second dose.
As I mentioned above, not holding back second doses is exactly what the Trump administration did, starting about two weeks ago. That said, I suppose it's (still) up to the states or localities to either keep second doses in reserve, or hope for timely additional shipments. And if you don't mind my saying so, from where I sit it's hard to judge the logic of your assumption about your local hospital's policy...
