Some people died in a panicked stampede after police employed tear gas, rubber bullets and baton charges, they said.
Thousands had gathered for a religious festival in Bishoftu, 40km (25 miles) from the capital Addis Ababa.
Some reports said police responded after anti-government protesters threw stones and bottles.
Others said demonstrators were entirely peaceful.
Ethiopia's government said 52 people were killed in the stampede.
Prime Minister Hailemariam Desalegn blamed "evil forces" for the deaths in a national address on state TV.