The Shulchan Aruch and Rama discuss starting Shemoneh Esrei along with a chazan in two different contexts. In one case, the tzibbur is davening normally, but a latecomer is ready to start Shemoneh Esrei when chazarat hashatz is about to begin.[1] A second case is one in which, under extenuating circumstances (e.g., the end time for davening is approaching), the tzibbur wants to recite heiche kedusha.[2] (In this context, we will not discuss the question of in which cases, if any, a congregation should choose the heiche kedusha system.)
In the former case, regarding a latecomer, the Shulchan Aruch describes the individual as starting to recite Shemoneh Esrei along with the chazan. Rav Ovadya Yosef posits that this is the correct approach both regarding the case of the individual and that of heiche kedusha.[3] Even though one should normally not respond to Kedusha if he is in the midst of his silent Shemoneh Esrei, he may do so in this case because he is reciting Kedusha in its correct place along with the tzibbur.[4] In contrast, the Rama rules that an individual should preferably commence Shemoneh Esrei only after responding to Kedusha and the concluding beracha. However, the Mishna Berura[5] cites significant Ashkenazic opposition to the Rama’s view. These poskim say that it is acceptable for an individual to start with the chazan. It may even be preferable to do so, especially during Shacharit; if the latecomer waits to begin his Shemoneh Esrei he will likely end up reciting Kedusha at the problematic juncture of “Shira Chadasha…”
Regarding heiche kedusha, the Rama recommends beginning one’s Shemoneh Esrei along with the chazan, as long as at least one person delays starting his Shemoneh Esrei in order to answer “amen” to the berachot.[6] While this would seem to answer your question, the Rama is referring specifically to a case in which there is insufficient time to daven normally, and some commentaries understand that the situation does not allow the tzibbur enough time to start Shemoneh Esrei after Kedusha. In any other case, they argue, it would be proper for the tzibbur to wait.[7] This answers your question in the opposite direction. Unless it will be impossible to finish Shemoneh Esrei on time, the tzibbur should begin Shemoneh Esrei only after Kedusha.
It is unclear why these commentaries conclude that starting along with the chazan is not preferable in the case of heiche kedusha. It is possible that they are concerned that there should be people to answer amen to the berachot. This is more of an issue for a tzibbur than for an individual.[8] Alternatively, the problem may be reciting Kedusha while in the midst of one’s Shemoneh Esrei.[9]
In fact, there are significant reasons to argue that it is preferable for the tzibbur to start Shemoneh Esrei together with the chazan. The Kaf HaChayim points out that if everyone were to listen to the chazan without davening themselves, it would appear as though he were reciting chazarat hashatz for them. This is problematic, the Kaf HaChayim[10] argues, because chazarat hashatz should not be recited before the tzibbur has said the silent Shemoneh Esrei. Rav Hershel Schachter posits that it is appropriate to say Kedusha only in the appointed place within one’s Shemoneh Esrei. During chazarat hashatz, it is as if the whole tzibbur is in the midst of Shemoneh Esrei; consequently, Kedusha is in its appointed place. However, in the case of heiche kedusha, in which everyone is reciting his own Shemoneh Esrei, if Kedusha is recited before people have begun Shemoneh Esrei, Kedusha is not in the right place.[11] Rav Schachter reports that Rav Soloveitchik had the practice of beginning Shemoneh Esrei with the chazan in the case of heiche kedusha.
Both positions on your question have significant support, and the stakes seem low, as most of the participants in the debate seem to disagree only as to which approach is preferable. Nevertheless, the majority opinion and the more prevalent practice among Ashkenazim is to wait until after Kedusha before starting Shemoneh Esrei during heiche kedusha (at least for Mincha, when heiche kedusha is more common). Those who follow Rav Soloveitchik’s rulings should start along with the chazan.