There are certain cases in which a lay person may give a blessing. The general introduction to the Book of Blessings, n. 18 states: “Other laymen and laywomen, in virtue of the universal priesthood, a dignity they possess because of their baptism and confirmation, may celebrate certain blessings, as indicated in the respective orders of blessings, by use of the rites and formularies designated for a lay minister.”
In the rubrics for the distribution of Holy Communion to the sick by extraordinary ministers, the minister may not bless the sick person but only may make the sign of the cross over himself and ask for a blessing upon both of them.
Based on these statements, the practice of extraordinary ministers giving blessings during Holy Communion does not appear to be in keeping with Church law.
Perhaps your priest should speak with the Extraordinary Ministers about the particular Church norms concerning extraordinary ministers of Holy Communion. Especially since “the canonical discipline concerning extraordinary ministers of Holy Communion must be correctly applied so as to avoid generating confusion.” (*Collaboration of the Non-Ordained Faithful in the Sacred Ministry of Priest * Article 8)
" Such norms should provide, amongst other things, for matters such as the instruction in eucharistic doctrine of those chosen to be extraordinary ministers of Holy Communion, the meaning of the service they provide, the rubrics to be observed, the reverence to be shown for such an august Sacrament and instruction concerning the discipline on admission to Holy Communion." (Collaboration of the Non-Ordained Faithful in the Sacred Ministry of Priest Article 8).
The answer to why people desire to receive the sacred species from the hands of a priest is contained in the document *Dominicae Cenae * #11 (last two paragraphs).