The Shulchan Aruch (YD 248:1) says that even a poor person who receives Tzedoka must give Tzedoka from the money he receives. The Shach says that this Din only applies if the poor person has enough to live on even after he separates the Tzedoka. However if he doesn't have enough to live on he is not obligated to give Tzedoka.
The Yad Avrohom says (YD 248) that a poor person who doesn't have enough to live on is only Patur from giving large sums to Tzedoka, however he is not completely Patur from Tzedoka and has to give a small amount of Tzedoka in order to be Yotzei the Mitzvah of Tzedoka.
Important Note: We try to convey the Tshuva to the best of our ability. We admit that our understanding may not be accurate. Please also understand that this Tshuva may not be the final word on this topic. One should consult a Rav before drawing any conclusions.