On private property, which covers your driveway and the building lot itself, the City does not require a permit for the container. That is also where Hutto's own rules expect it to be, since the jobsite containment requirement assumes the container is on the site.
The right-of-way is a managed process rather than a simple counter permit. The City has adopted a right-of-way management ordinance covering all users of the City's right-of-way, along with a permitting and construction manual, separate registrations for service providers and for right-of-way contractors, insurance and bonding requirements, and its own fee schedule. All of that is published on the City's right-of-way management page, which notes the material is still being developed. Nothing in the City's code bars or permits a container in the street by name, so a street placement is a question to put to the City before you book rather than an application you can simply file.
The jobsite rule is the one that matters most on a build. The City requires construction debris on all building and construction jobsites inside the city limits to be contained in a dumpster, a roll-off container, or a wire mesh fenced area with a mesh too fine for a four-inch sphere to pass through, and it also allows a temporary container built from half-inch oriented strand board. Whichever route you use, the owner, the contractor and everyone on site has to take all reasonable measures to stop debris blowing out. The same conditions require a clear, accessible path on the jobsite that construction debris and equipment do not block, for the whole permit and construction period.
Being straight about it: the mesh fence is a cheaper way to comply. It also leaves you handling the debris twice and does not solve the haul. A container satisfies the rule and removes the material in one step. For the City's number, call 512-759-4015, and your account manager can help you work out which route suits your site.