Unite and the Royal College of Nursing said that although some school nurses receive training on the issue, many don't.
There are currently around 1,500 school nurses in England, but there is no statutory requirement for them to undergo training in identifying and supporting young people with potential mental health problems.
Rosalind Godson, professional officer of the health sector from Unite, said mental health training should be statutory, and incorporated into all school nurses’ training.
“Mental health issues are so common that all school nurses should be formally trained to be able to recognise them and offer help and support in the early stages to prevent worsening.
"However they also need training in order to recognise when the problem is serious and needs referral."
Godson says training for school nurses has “always been patchy”.
“In some areas, nurses are trained in mental health, but other areas don’t do that,” she said.
She said it is likely that some nurses have had local training in the form of study days, a few have had more formal training, and many have learned on the job.
Ian Hulatt, professional lead for mental health nursing at the Royal College of Nursing said there should be standardised mental health training for school nurses across the country.
“It is important for school nurses to be enabled, as part of a continued professional development by employers, in child mental health,” he said.
The government's Future in Mind report, in its response to the findings of the Children and Young People's Mental Health Taskforce, states that school nurses can play a crucial role in supporting the emotional and mental health needs of school-aged children.
"School nursing services are universal and young people see them as non-stigmatising," the report states.
However it does not outline any plans to improve the training they receive or set out what they should be doing.
The call for improved training coincides with government efforts to improve mental health services for young people. In March this year it announced an investment of more than £1bn to improve services.
However Godson said the government drive to improve young people's mental health is contradicted by the strong focus in the school system on exams, which can increase stress and anxiety.