Teachtaireacht is an indefinite nominative singular feminine noun, so there’s no need to lenite it. Pearsanta is an attributive adjective that applies to a non-genitive singular feminine noun, so it needs to be lenited.
Scilling is right (to whoever down voted him). Teachtaireacht is a feminine noun, which lenites (puts a h on) the adjective following it. (In this case, although Pearsanta is a noun, it is being used as an adjective).