The First Minister was told that families affected by "failures in the system" of mental health services should not be failed again by the independent inquiry.
Scottish Labour leader Richard Leonard asked Ms Sturgeon, at First Minister's Questions, if she would reassure affected families, and also ensure they had an advanced copy of the inquiry's report.
He said he had been told by Gillian Murray, who's uncle David Ramsay took his own life after being turned away by the Carseview Centre and who had campaigned for the inquiry, that "nothing seems to have happened, we're not kept involved, it's definitely not transparent."
To a silent Holyrood chamber, Mr Leonard said that the voices of "people like Mandy McLaren, the mum of Dale Thomson who tragically completed suicide in 2015" - also after leaving the Carseview Centre at Ninewells Hospital - "must be heard".
He added: "The terms of reference set for this inquiry state that it must consider the perspective and give voice to families patients, carers and others who have experience of suicide or involvement of mental health services within Tayside.
"Will you ensure that the families receive an advanced copy of the interim report, which is due in the next few weeks, will you listen to voices of families and do what you can do to help restore their confidence in this inquiry?"
The independent inquiry into NHS Tayside's mental health services was launched last September, headed by David Strang, former prison's inspector. Anyone who has engaged with mental health services in the region was urged to share their experiences.
An investigation was initially ordered into Dundee's Carseview Centre but was expanded following a campaign by families of people who took their own lives. The inquiry is expected to review safety, care standards and access to mental health services.
Mr Leonard raised the issue as the Scottish Parliament marked Mental Health Awareness Week. He said: "As a society we're increasingly open about and understanding of mental health, but we need to do much more to get our mental health services right, especially at the point of crisis."
Ms Sturgeon said that she "would want to learn lessons from the experiences of the kind narrated by Richard Leonard and my sympathies are with any family has had an experience of that nature."
She added: "We established an independent inquiry into mental health across Tayside as a result of some of the cases he [Leonard] brought to chamber. That inquiry hasn't reported yet but when it is, it will be fully scrutinised by this chamber and the government and the health service more generally. I think it would be wrong to pre-empt the outcome of that inquiry but I will give an assurance we will take forward any recommendations it makes.
"I know David Strang has met with family members and that would have been expected of him, and it would be my full expectation that an advanced copy of the report would go to those affected directly. I would stress again though that it's an independent inquiry and it's right the government allows it to be conducted entirely independently."
She added: "It's important we continue to tackle stigma [of mental health] and invest in preventative measures and invest in specialist care when people need it.
Scottish Liberal Democrat leader later also pressed the First Minister on mental health, telling her a GP had told him "he had stopped referring patients to mental health services because the waiting times are so long or there is no prospect of them every getting treated. The FM has promised those patients that they will get treatment when they need it. They feel let down, are they wrong?"
Ms Sturgeon said that patients would rightly feel "aggrieved" at waiting too long and that she "would apologise to anyone in that situation", but she pointed to an investment of £850m to meet waiting times targets and "in terms of mental health we're investing significantly to improve services, not just specialist but preventative and community services as well."
Mr Rennie replies that if "warm words could treat people faster then thousands wouldn't be waiting." He said one in five people are waiting over 18 weeks, some waiting as long as two years, and others receiving no help.
"The First Minister says she takes this seriously but her government's mental health strategy was 15 months late, the suicide strategy was 20 months late and is 700 staff short of it's own recruitment plan. GPs, A&E department and police officers have to pick up the pieces because these patients have nowhere else to go. In mental health week, years, years after I first asked her, why are people still waiting so long?"
The First Minister rejected the suggestion that all she was offering were warm and said: "We're seeing budgets increasing, now over £1bn and staffing number rising and with the mental health and suicide strategies we took more time to make sure we took views of all stakeholders into account.
"We want fewer people to need specialist services because of investment in preventative and community services and that's where our focus lies."