On private property, including your driveway or a fenced job site, no City permit is needed for the container itself. The street is a different matter, and in Bountiful it is harder than in most towns.
The City's code enforcement page sets it out under blocking sidewalks, roads, and park strips: nothing may be placed on a public street, sidewalk, or right of way that obstructs it or the free passage over it without the permission of the governing body. Permission from the governing body means the City Council, not a counter clerk, so this is not a same-week errand. Blocking a traffic lane is stricter again: that needs the Police Chief's permission, applied for in writing at least five days ahead, with the written consent of every property owner fronting the blocked stretch or within 200 feet of it.
For a building project the question answers itself. The City requires the contractor or owner-builder of every residential or commercial construction site to keep a commercial trash bin, and a portable toilet, on the building lot and not on a street, sidewalk, or other public property, from the first day of construction through the last, and to have the bin emptied when it fills. Only the City Building Inspector can waive that, in writing, and only where equivalent facilities are already close by. So plan on space in the driveway or on the lot.
Call the Bountiful Engineering Department at 801-298-6125 if your site plan has no room away from the street, or the Planning Department at 801-298-6190 about yard and screening rules. City offices run Monday through Thursday, 7 a.m. to 6 p.m., so a Friday question waits until Monday. Your account manager can help you sort out which office you need before a delivery date is set.