The head of the church, Rev Trevor Gribben, stepped down on Wednesday after "serious and significant failings" in safeguarding between 2009 and 2022.
During part of that period, he had a senior role in the church as general secretary.
Speaking as a church member, Gordon Lyons said: "I don't think we have been furnished with all the information.
"A lot of people will be concerned at what has been said and also be concerned about what hasn't been said."
He told Good Morning Ulster it was difficult to make any judgement with so little information and called for more clarity on what exactly had happened.
"Our thoughts are with those who are affected. This is a serious issue and we need to make sure people are protected and are safe," he added.
The minister said he hoped people would come forward.
'Damage limitation'
The lack of information contained in the announcement on Wednesday has come under some criticism.
Lord John Alderdice, who resigned as a Presbyterian elder in 2018, accused the church leadership of not doing enough.
He also was the co-author of a report alleging a culture of bullying and abuse of power in the church.
"What you saw yesterday was not openness and transparency. It was a damage limitation exercise," he told Good Morning Ulster on Thursday.
"It was an attempt to smooth over what cannot be smoothed over any longer. There are massive institutional failings."
Lord Alderdice said he has been "concerned for some time" about safeguarding.
What have Presbyterian ministers said?
Presbyterian minister Reverend Cheryl Meban told BBC Radio Ulster's Talkback programme that she is "deeply sad" about the news.
"As a church, we do need to recognise that there is a culture of control, in any institution and this is not just churches.
"To see people that I know, have love and respect for, taking a rap is difficult and heartbreaking, but at the same time it's nothing like the suffering of people who have been shut down and left to feel that their story is not valued and important enough to be taken seriously, which happens in all social groupings not just the church," she added.
"But I am part of this church and so I do share the blame and share the responsibility for having kept an institution going that really has hurt people."
She said that any a review should be external.
"No institution can be trusted to police itself," she added.
"Trying to manage this as the church they are still potentially perpetuating a sort of abusive control that is hurtful and demeaning to victims."
PSNI investigation
On Wednesday, the acting Clerk of the General Assembly, Rev Dr David Allen, said the church had initiated an investigation after receiving information from the PSNI in May.
"We became aware of the actions of a member of one of our congregations – not within that congregation, but in another area of his life.
"The PSNI told us they were investigating this individual.
"They prevented us from taking any action until their investigation was completed. When this investigation was complete, they gave us the go-ahead to carry out internal investigations into our processes."
He said there were three Presbyterian congregations "where we are aware of people who have come forward and spoken with us".
He said congregations on both sides of the Irish border had been affected.
How many cases is the church aware of?
The general council convener Rev David Bruce said the church does not know "precisely who has been harmed".
Rev Bruce said the number of cases they are aware of is "very small," however he said the church was keen to hear from anyone with concerns.
Rev Bruce said the church had instituted a response line because "we want people to get in touch with their stories".
"It is inevitable that there are people out there, whether they are younger or older who have been affected by this and the important thing for us is we can hear from them, respond to them and channel them towards the right sorts of support they will need," he added.
When put to him that the church does not know the number of cases because records were not kept and individual situations were not appropriately dealt with, Rev Bruce said "that is correct".
He said it is "without excuse".
"All organisations face a perpetual challenge, whether in secular or religious world or whatever, in managing their culture," Rev Bruce said.
He said it is "easy for an institution to become culturally lazy where it is assumed that certain things are happening because people talk a good game or because we got into the habit of believing that certain things are happening".
Rev Bruce said the church has committed to improving its oversight and supervision "of all of these matters".
He said it will also ensure there is "external supervision of these activities from people with professional training, but who are at arm's length from us, so it's not us marking our own homework but we have people looking in from the outside and assessing how we are doing".
Will the church publish the report?
Asked if the Presbyterian Church would publish the report, Rev Bruce said the church would "have to take legal advice" because "it involves some individual people ... who have been affected adversely by this".
The Charity Commission said it was currently reviewing the information provided by the charity under the serious incident reporting process and is "engaging with the trustees to ensure that appropriate steps are taken to protect beneficiaries and restore confidence".
It has asked for a copy of the most recent internal investigation carried out by the church and it has been engaging with the church over a "separate dossier" received.
"The commission will examine whether there are any overlapping issues between the matters raised in the dossier, those highlighted in the statement issued yesterday and within the investigation report, once received from the charity," a spokeswoman said.
Rev Trevor Gribben, who was formally installed in the position in June, will also be leaving his roles as clerk of the assembly and general secretary.
The Moderator for the Presbyterian Church in Ireland is the most senior office-bearer in the organisation.