During the summer months, many congregations accept Shabbat early on Friday afternoon, and recite Minha and Arbit on Friday evening before sundown. This arrangement gives rise to a number of different questions concerning the recitation of Minha and Arbit.
One such question concerns a person who arrives late in the synagogue on Friday evening, after the congregation has already accepted Shabbat through the recitation of "Lecha Dodi." As the sun has not set, this individual still has the opportunity to recite Minha. Nevertheless, he may not recite Minha in the synagogue once the congregation has already accepted Shabbat. Since the congregation recites a Shabbat prayer service and has already accepted the Kedusha (sanctity) of Shabbat, it is inappropriate for somebody to recite a weekday prayer in the synagogue at the same time. He must therefore recite Minha in the hallway or in a separate room. After the congregation completed the prayer service, however, one may certainly recite Minha in the synagogue. Similarly, another Minyan can then pray Minha and Arbit in the same room. The Halacha mentioned above pertains only to reciting the Minha at the same time when the congregation recites Arbit.
Furthermore, this Halacha applies only on Friday evening, and not on weekdays. If on a weekday a person comes to the synagogue when the congregation recites Arbit, and he wishes to recite Minha (assuming, of course, that the sun had not set), he may do so. Since both the individual and the congregation recite a weekday prayer, it is permissible to recite Minha as the congregation recites Arbit.
A number of questions arise in the reverse situation, as well, when a person has already recited Arbit and now encounters a congregation reciting Minha. This often happens in places like the Western Wall or very large synagogues where multiple Minyanim are held simultaneously. Hacham Ovadia Yosef rules that in such a case one should respond to Kedusha and answer "Amen" to the Berachot he hears. Even though he had already recited Arbit, there is no Halacha forbidding answering "Amen" or responding to Kedusha of Minha.
Another question was asked concerning a person who recited Arbit and is now asked by nine people to join them in making a Minyan for Minha. Regarding this case, Hacham Ovadia Yosef draws a distinction between weekdays and Shabbat. On weekdays, a person who had already recited Arbit may, indeed, be counted toward a Minyan for Minha. On Friday evening, however, a person who had already accepted Shabbat cannot count toward a Minyan for the weekday Minha. Since he has already begun Shabbat, he cannot participate in forming a Minyan for a weekday prayer service.
(These Halachot are taken from Hazon Ovadia – Laws of Shabbat, p. 361 and onward.)
Summary: One may not recite Minha on Friday afternoon in the synagogue at the same time as the congregation recites the Friday night Arbit; he must go out into the hallway or another room. On a weekday, however, one may recite Minha in the synagogue while the congregation recites Arbit. One who had already recited Arbit and hears a Minyan reciting Minha should respond to Kedusha and answer "Amen" to the Berachot. Likewise, one who recited Arbit can count toward a Minyan for Minha, except on Friday night, when a person who already accepted Shabbat cannot be counted toward a Minyan for the weekday Minha.