Normally on Rosh Hodesh, the Shaliah Tzibbur recites the blessing for the Hallel on the congregation’s behalf and has the intention to acquit the congregants off their obligation. Rabbi Shalom Messas (Tevuot Shamesh § 68) discusses whether an an individual may make a blessing for oneself, if it causes one to pray with more inspiration, for example, and rules that this would constitute an unnecessary blessing. Rabbi Machlouf Abuhatzira (Yetze Sha’a § 14) writes that if the Shaliah Tzibbur is more distinguished than the individual, then the former should make the blessing, otherwise one may make one’s own blessing, and Rabbi Shalom Messas agrees with this stipulation. Furthermore, the Meiri notes that even if one follows the opinion of the Rif (only the Shaliah Tzibbur makes a blessing) and arrived at the synagogue while the congregation was already saying Hallel, one may make a blessing individually.
Regarding the concluding blessing of the Hallel, “Yehalelucha”, Rabbi Shalom Messas rules that similar to the opening blessing, an individual who missed hearing it from the Shaliah Tzibbur may recite in oneself.
Summary: One may recite the opening and closing blessings of the Hallel on Rosh Hodesh individually, so long as one is praying with a Minyan.