On private property, including your own driveway or a job site you control, Beekman publishes no container permit at all. For most rentals here, that is the whole answer.
The Town of Beekman publishes a complete list of what its Building Department issues permits for: new construction, additions, alterations, pools, decks, sheds, heating and cooling work, plumbing, electrical work, fuel tanks and signs. A dumpster is not on that list, and no separate street occupancy permit appears anywhere on the Town's published pages.
Roadside placement is the one case worth a call. The Town's Highway Department handles driveway permits and the roads it maintains, and it asks residents to phone before starting work in the right of way. You can reach the Highway Department at 845-724-5300, extension 209 or 210. Beekman Town Hall is at 4 Main Street in Poughquag, open Monday through Thursday, 8 a.m. to 4:45 p.m., and closed Fridays.
Some roads through the hamlet are county or state routes rather than town roads, which changes who you ask. Your account manager will help you work out which office covers your stretch before delivery day.