The monitoring group said the rare visit to Germany by Turkmenistan's reclusive president Gurbanguly Berdymukhamedov on Monday should be used to tackle the plight of activists.
"Chancellor Angela Merkel should not waste this opportunity to directly and insistently call for oppression in Turkmenistan to be ended," said HRW's director for Europe and Central Asia, Hugh Williamson.
Dozens of people were arrested in the late 1990s and early 2000s and vanished in the country's prison system, and their families received no official word on their whereabouts or wellbeing, he said.
HRW highlighted the disappearances of opposition leader Boris Shikhmuradov, his brother Konstantin and former OSCE ambassador Batyr Berdiev as key cases to address.
It also referred to local reporter Saparmamed Nepeskuliev, who has been incarcerated for almost a year on narcotics charges.
"Chancellor Merkel should press Berdymukhamedov for Nepeskuliev's release," Williamson said.
"He is sitting in prison for doing what correspondents around the world do: he shares information about what is happening in his country."
Germany should urge Turkmenistan to submit a new draft constitution that would include no term limits for the president, allow censorship and offer no protection of the freedom of movement of its citizens to international review, HRW said.
Williamson noted that Turkmenistan had already imposed a travel ban on many dissidents, a policy he said should make Merkel particularly wary given that she grew up in communist East Germany which had similar restrictions.
Rights groups regularly list ex-Soviet Turkmenistan, which has the world's fourth largest known reserves of natural gas, as one of the least free states.