Answer: The Gemara (Berachos 9b) teaches that one should not interrupt between the beracha of geula, redemption, and the amida. This applies during shacharis and maariv.
The Shulchan Aruch (OC 236:2) writes that the chazan announces ‘rosh chodesh’ before shemoneh esrei to remind everyone to say yaaleh veyavo. As this is necessary for davening, it is not considered to be an unnecessary hefsek (See Shut Harashba 1:293). However, the Mishna Berura (Shaar Hatziyun 236:4) writes that this only applies during maariv. One may not announce this during shacharis, when there must not be the slightest interruption before the amida (See Taz OC 114:2).
The Kaf Hachaim (OC 236:16) writes that the chazan may announce the words ‘yaaleh veyavo’ loudly while saying his amida, even starting his amida early if necessary. R’ Shlomo Zalman Auerbach (Halichos Shlomo, Rosh Chodesh 1:1) maintained that it is inappropriate for anyone other than the chazan or gabbai to do so then. One is allowed to hint to someone else to say yaaleh veyavo, however. Similarly, R' Moshe Stern (Baer Moshe 4:10) writes that while anyone may say the words ‘yaaleh veyavo’ loudly, no one else should do so afterwards.
The Kaf Hachaim (OC 236:17) notes that in Yerushalayim, the minhag is not to announce anything. Elsewhere one may announce ‘yaaleh veyavo’ during maariv, though not other announcements that are of lesser importance, such as al hanissim. Nonetheless, the Mishna Berura (236:7) allows all such announcements (See Magen Avraham 236:1).
In conclusion, the gabbai may bang on the table before the amida in shacharis¸ and may say the first couple of words of yaaleh veyavo out loud during his amida, though others should not do so. In many shuls the gabbai announces ‘yaaleh veyavo’ before the amida of (mincha and) maariv.