Reuters reported that Indonesian security forces responded to a riot - where they opened fire on a crowd - late last month.
The riot began after locals started throwing rocks at Wamena police station in Highland Papua where a man accused of abducting a six-year-old was detained.
Catherine Delahunty, a spokesperson for the network, said the group condemned the decision to open fire.
"The context of the event may be debated but the reaction of the Indonesian government which is constantly to shoot first and protect their military from any consequences of their actions is unacceptable.
"This is an unacceptable, continual breach of civil rights, that civilians can be shot dead in a street incident without warning. And there are no consequences. This is the definition of a breach of human rights.
"It was raised in Geneva last year at the Universal Periodic Review on Indonesia by a number of countries and organisations, and yet it's still happening. People are still being killed in this manner."
Delahunty wants New Zealand's Foreign Minister, Nanaia Mahuta, to speak to Jakarta about such situations.
However, the incident comes as Kiwi pilot Philip Mehrtens remains a hostage of the West Papua National Liberation Army and Delahunty acknowledged that it was a difficult time for the government in Wellington because they are focused on negotiations to free Mehrtens.
However, she said the ongoing situations cannot be ignored.
"There are battalions of military being moved into the Highlands because of situations like Wamena. There is no accountability of the military. It is a very dangerous situation."
RNZ Pacific has requested a response to Free West Papua Aotearoa's position from the Indonesian Embassy in Wellington.
Photo: Supplied Caption: Arrival of more Indonesian troops in Jayapura, Papua, March 2021