Eight vigilantes were killed in the first blast in the Igwanda area of the city at about 12:15 pm (1115 GMT) while two died in the second outside a tavern and brothel in the Dorawa area.
There was no immediate claim of responsibility but Potiskum, the commercial capital of Yobe state, has been hit repeatedly by Boko Haram in its six-year insurgency, including by suicide bombers.
The first attack targeted the civilian vigilante office in the Igwanda area of the city and came after a man seen carrying laptops and other computer accessories was detained.
"But we didn't know he had explosives on him. He detonated the explosives and we lost eight men, including our commander," said the vigilante, who asked not to be named for his own safety.
A police source in Potiskum confirmed the attack. "We took eight dead bodies of the vigilante men to the hospital and the remains of the suicide bomber."
The second blast happened moments later in the Dorawa area of Potiskum at an open air tavern and brothel that has previously been hit by the Islamists.
"Two people have been confirmed dead," he added.
Boko Haram are keen to prove they are not a spent force despite a four-nation military offensive that has run them out of captured towns and villages since February this year.
The use of improvised explosive devices and suicide bombings has increased since May 29, when President Muhammadu Buhari took office vowing to crush the militants.
At least 162 people have been killed in some 17 separate attacks in that period, according to AFP reporting.
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