Is it true that we should give our Zakaat locally?

Yes, according to hadith of Mu`az r.a. of Bukhari-please read Tous sur les Zakaat on web- we should give our Zakaat firstly to those in need in our family. The holy prophet SAW said that the Zakaat given to the family will have 2 rewards-Tirmizi. Some fuqahas said that if you have poor relatives, then your zakaat will not be accepted elsewhere. You will have to give your relatives. Furthermore, it is makrooh that we take our Zakaat out of the country because each wealth has to be distributed to its poor to its area.

Shami:

(و) كره (نقلها إلا إلى قرابة) بل في الظهيرية: لا تقبل صدقة الرجل وقرابته محاويج حتى يبدأ بهم فيسد حاجتهم (أو أحوج) أو أصلح أو أورع أو أنفع للمسلمين