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The Kashrus status of purchased Tzitzis:

Purchasing from a Jew:[1] One who purchases a pair of Tzitzis from a Jew can assume that the Tzitzis is Kosher [without doing any further inquiry].[2] [Nevertheless, in today’s times one can no longer rely on an unfamiliar Jew, even if he is observant, as Halachic mistakes and forgery are very common.[3] Thus, one is to only buy Tzitzis with a reliable Hashgacha.]

Purchasing from a gentile:[4] One who purchases a pair of Tzitzis from a gentile merchant, and the gentile states that he purchased the Tallis together with its Tzitzis from a Jew, one can assume that the Tzitzis is Kosher.[5] This applies even if the gentile’s main business is not in selling Tzitzis, but in selling other merchandise. However, this only applies if one purchased a Tallis with Tzitzis that are properly tied to its corners, however plain Tzitzis strings [without a double Chosem[6]] that are not attached to a Tallis may not be purchased from a gentile, even if he is a merchant, as perhaps he made them himself.[7]

Found Tzitzis in the open:[8] One who finds Tzitzis strings in the marketplace, the Tzitzis can be assumed to be Kosher if they are interwoven and cut into four long strings, as certainly they have been made for the sake of Tzitzis.

Summary:

All Tzitzis that are purchased from a Jew, or found in the open, are assumed to be Kosher. [In today’s times however one can no longer rely on an unfamiliar Jew, even if he is Torah observant, as Halachic mistakes and forgery are very common. Thus, one is to only buy Tzitzis with a reliable Hashgacha. The same applies regarding one who finds Tzitzis strings, that they should not be used unless they contain a reliable Hashgacha.] One may not purchase Tzitzis strings from a gentile unless they [contain two signs and a Hashgacha, or contain a Hashgacha and] are already properly tied to the Tallis.