Many police and NHS workers are too concerned with battling the pandemic to look after their mental health, they were told.
Insp Phil Spencer from Cleveland Police said staff did not engage enough with counselling "because we don't want to take anybody else's valuable time".
Prince William said he "really worries" about the effect on front-line workers.
"When you're surrounded by that level of intense trauma and sadness and bereavement, it really does, it stays with you at home, it stays with you for weeks on end," he said.
Insp Spencer said emergency workers "run towards danger, run towards a terrorist attack, we run towards the pandemic".
"Perhaps further down the line when all this is gone we're going to have some broken police officers and emergency services staff, because we're too busy focusing on protecting the most vulnerable," he said.
The couple also spoke to counsellors from Hospice UK's Harrogate-based Just B support line for NHS staff, social care workers, carers and emergency services, which their foundation helps financially.
The prince said he feared "you're all so busy caring for everyone else that you won't take enough time to care for yourselves".
He and Catherine said the stigma surrounding seeking help for mental health issues must end.
Individual rights are not subject to a public vote; a majority has no right to vote away the rights of a minority.