According to IRS Minister Audit Technique Guide, “Contributions made to or for the support of individual missionaries to further the objectives of their missions are includible in gross income (Rev. Rul. 68-67, 1968-1 C.B. 38)”. The Minister Audit Technique Guide also gives an example of a court case that discusses this topic, "In the Charles E Banks and Rose M Banks v. Commissioner, T.C. Memo. 1991-641, case a structured and organized transfer of cash from members of the church took place on four special days of each year. Prior to making the transfers, members of the Church met amongst themselves to discuss the transfers. The amounts of the transfers were significant. Several members testified in Court. Their testimony indicated 'the primary reason for the transfers at issue was not detached and disinterested generosity, but rather, the church members' desire to reward petitioner for her services as a pastor and their desire that she remain in that capacity.' The Court ruled the transfers were compensation for services hence included in gross income."
The "specific individuals" referenced in the question above should be contributing their support through a church or other Internal Revenue Code section 501(c)(3) organization such as a mission agency. These organizations are "in the business" of engaging missionaries to perform gospel work in their appointed location and duly report their support as taxable compensation.
Only contributions made to a church or recognized not-for-profit organization are tax deductible by donors. Gifts by individuals to other individuals for whom no services are provided are non-taxable gifts to the recipients.
Ministers by definition of the Internal Revenue Code are self-employed for purposes of the federal self-employment tax. Missionaries solely supported through mission agencies typically are provided either Form 1099-MISC (if the agency classifies its missionaries as independent contractors) or Form W-2 (if the agency classifies its missionaries as employees). In both cases, agencies typically do not withhold taxes since ministerial earnings are exempt from mandatory withholding. Some agencies, however, do provide for elective withholding of federal and state income taxes (but not FICA taxes) for the convenience of their clients who would otherwise be required to make estimated tax payments.
See an earlier post regarding this complex topic: