Answer: The Mishna Berura (146:15) writes that there is a machlokes as to whether the tzibbur should normally read along with the baal korei. According to the Shelah and Gra, one should follow without reading, while the Mateh Moshe writes that by reading along quietly one will be able to concentrate better.
The Shulchan Aruch (OC 141:2) writes that the one called up to the Torah must read along as otherwise he would be saying his berachos levatala.
The Taz (OC 141:2) challenges why the one receiving the aliya can’t say the beracha on the baal korei’s behalf as we have a rule (Gemara Berachos 21b; Sukka 38b) shomea keoneh, that one can fulfil one’s obligation by listening. Thus, the Gra (OC 139:3) disagrees, writing that the one called up for his aliya should not read along quietly. R’ Moshe Sternbuch (Teshuvos Vehanhagos 1:399) writes, however, that we pasken like the Shulchan Aruch, and one called up for an aliya must read along.
As listening to parshas zachor is mideoraisa, there is a machlokes as to whether this applies.
R’ Chaim Elazar Shapira (Minchas Elazar 2:1:10) writes that as there is a mitzva mideoraisa to remember amalek, everyone should read along parshas zachor quietly with the baal korei rather than relying on shomea keoneh.
R’ Ovadia Yosef (Yechave Daas 3:53), however, quotes the Pri Chadash (OC 67:1) and the Netziv (Meshiv Davar OC 47:128) who maintain that the tzibbur should follow along quietly (See Piskei Teshuvos 685:7).
This is so important that according to some poskim even the one called up for the aliya of parshas zachor should remain silent. Thus, R’ Ephraim Greenblatt (Rivevos Epraim 7:344) writes that when R’ Moshe Feinstein was called up for parshas zachor, he would listen quietly to the baal korei. R’ Moshe Sternbuch explains that the reason for reading parshas zachor is different to the regular leining where the reason is for the tzibbur to hear the parsha. Thus, while according to the Shulchan Aruch, remaining quiet would usually constitute a beracha levatala, here it wouldn’t, as he is being yotze with shomea keoneh just like everybody else.
Nonetheless, R’ Ephraim Greenblatt (Rivevos Ephraim 1:447) writes that it is ideal for the one who is called up to read along quietly with the ball korei.
In conclusion, the tzibbur should be quiet while parshas zachor is being read. In some shuls even the one receiving the aliya doesn’t read along with the baal korei.