His lawyers say the case is connected to Titiyev's investigations into the secretive prison system of a republic that is notorious for rights abuses.
"We ask that Titiyev be found guilty and sentenced to four years in a penal colony, along with a fine of 100,000 rubles (USD 1,500)," prosecutors told the court in Chechnya's Shali district, according to Russian news agencies.
A verdict in the trial, which has been ongoing since last summer, is expected shortly.
Observers say drug charges have been used repeatedly in the past to silence critical journalists and rights workers in the region.
The NGO said its offices in the neighbouring Ingushetia republic were torched following Titiyev's arrest and colleagues in Dagestan, another Caucasus republic, were attacked.
Titiyev's predecessor as the regional leader of Memorial, Natalya Estemirova, was kidnapped and killed in 2009.
The rights group said the killers were acting on the orders of local authorities.
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