A nurses' union has urged Stormont ministers to provide more clarity on the supply of personal protection equipment (PPE) and tests.
The Royal College of Nursing (RCN) in Northern Ireland has written to ministers voicing its concerns.
A GP and a social worker have also spoken to BBC News NI about the situation.
Health Minister Robin Swann said that "every conceivable effort" was being made, but there was no "quick fix".
RCN Director Pat Cullen said its members were very concerned that they are not getting access to the PPE they need.
She said: "We must, as a matter of urgency, get absolute clarity about how much stock we have, where it is, how is it being distributed and how much is definitely on the way.
"We will no longer accept promises - we must now deal in facts."
Ms Cullen also voiced concern that testing kits made in Northern Ireland were being sent to the rest of the UK.
Meanwhile, a County Fermanagh GP has said that doctors are “very concerned” about the lack of PPE available.
Dr John Porteous said health staff “need to know more is coming”.
Mr Porteous works in Lisnaskea at a practice with nine doctors and more than 14,000 patients.
He said his surgery was given just 12 PPE packs a few weeks ago, and a further 12 packs last week.
“We obviously have a small number of these kits so we have to conserve those for those higher risk consultations and, at this point, we would see that would be more likely to happen [during] home visiting when, maybe, an elderly person or someone who is not able to move needs to be seen and examined up close,” he said.
'Staff are a bit scared'
Mr Porteous said that health authorities asked for only two doctors from the practice to be put forward for specially-fitted masks, known as FFP 3 masks.
He added that his staff are "all feeling very uneasy and a bit scared".
One social worker, who spoke anonymously to BBC News NI, said that many of his colleagues were self-isolating, and one entire team has tested positive for Covid-19.
"We are dealing with intense domestic situations, and we just do not have the PPE to deal with those situations," he said.
"You are asking staff to risk themselves or buy their own.
Staff and public frustration
"Our entire team of around 300 social workers got three bags of PPE with each bag consisting of two masks, two sets of gloves, two aprons and a disposable bag, no hand sanitizer, and nothing else,” he said.
The social worker said staff “would have expected that multiplied by 100”.
He added: “I'm a father of children, I take every precaution I can, but my concern is you can only do so much. I'm concerned about passing this on to my family.”
While Health Minister Robin Swann has acknowledged the scale of staff and public frustration on the issues of testing and PPE, he said the Department of Health has faced “significant” challenges.
“The situation with PPE supply has been made more challenging by the fact that supply routes from China, a leading global provider, were closed until recently,” added Mr Swann.