PCCs are “operating with one arm tied behind their back”, according to the right-leaning Policy Exchange.
The move would mean cheaper and more effective justice, and better accountability, the think tank said.
The Home Office said it would look at new roles where there was potential to save money and improve services.
PCCs, who replaced existing police authorities in 41 force areas across England and Wales, apart from London, set force budgets and hire and fire chief constables.
But Policy Exchange, which claims to be the first organisation to propose the idea of PCCs, said they should also be able to carry out prison and police force inspections, and hire and fire prison governors, court managers and senior probation officials.
It said commissioners would in effect become “local ministers for policing and crime”.
Its report, Power Down, proposed a pilot scheme involving 10 PCCs.
London's deputy mayor for policing and crime should be among them but the rest could be drawn from any PCCs that want to take on the extra responsibilities, which would not attract any extra pay, it said.
Policy Exchange report author Max Chambers said: “Currently, PCCs are operating with one arm tied behind their back.
“They must be able to hold local criminal justice agencies to account. That includes the ability to appoint the right people as well as set the local strategy and hold criminal justice leaders to account for performance.”
He went on: “It will mean cheaper, more effective justice in a system that desperately needs an injection of dynamism and it's the best way of making sure that PCCs fulfil their potential and meet the promises they made to the electorate.”
Hertfordshire's PCC David Lloyd, who has pressed for more powers for PCCs since they were elected in November, backed the report.
“Ministers have been clear from the start that for maximum impact police and crime commissioners need to be big players in the criminal justice arena,” he said.
“It is time to follow through on these plans.”
Meanwhile, Northamptonshire's PCC Adam Simmonds said the report was “challenging us to lead, move further and push harder to get local communities more in control and providing local leadership of some key public services”.
A Home Office spokesman said: “Police reforms are working and crime is falling. The introduction of PCCs has been integral to these reforms by making forces more accountable to their communities.
“We welcome this report and will consider new roles for PCCs where there is the potential to deliver greater efficiencies, improve services and protect the public.”
The MPs' report said it seemed “very easy” for a PCC to remove a chief constable for “insubstantial” reasons – but the Home Office said strict protections were in place.