The incident is said to have taken place on Tuesday in Sichuan province's Ganzi prefecture, also known as Kardze.
Arrests were also made and some people fled, the activist groups said.
The incident does not appear to have been reported in Chinese state media.
Obtaining independent confirmation of events both in Tibet and in ethnic Tibetan areas in surrounding regions is extremely difficult.
Both access to these areas and information flow out of them is tightly controlled.
Chinese state media does confirm some of the incidents but not all. Accounts from activist groups have proved reliable in the past.
According to UK-based group Free Tibet, a village leader named Wangdak was arrested on Monday over a dispute with local authorities.
The group said the row related to alleged harassment of female members of a dance troupe at a celebration villagers had been ordered to stage for senior officials.
The US-based International Campaign for Tibet said it also related to a dispute over official restrictions on a traditional gathering at a local horse festival.
After Mr Wangdak was detained, a crowd of Tibetans gathered to protest.
Both groups said armed police were deployed, used tear gas and then opened fire.
Mr Wangdak's son was among those who were shot, both activist groups said.
Free Tibet said at least two people were shot but the nature and cause of the other injuries was not clear.
The village was now surrounded and many adults had gone into hiding, Radio Free Asia reported, citing a Tibetan exile monk.
The Tibet issue: Two views
Inside Tibet
Many ethnic Tibetans live in Gansu and Sichuan provinces, which lie near Tibet. Activists say China enforces tight restrictions over Tibetans' religious and cultural activities.
China argues its investment into Tibetan areas has greatly advanced standards of living.
In recent years more than 100 young Tibetans have set themselves on fire in what activists say are protests against Chinese rule. Most of these incidents have taken place in Tibetan communities outside Tibet.
There have also been other shootings. Last year, activist groups said Chinese police opened fire on Tibetans who had gathered to mark the Dalai Lama's birthday, injuring several.
Ten people were injured when Chinese police opened fire on Tibetan protesters demonstrating against the detention of a village leader, two activist groups and overseas news reports say.