When praying the Amidah at the second prayer of "Atá Gibor..." during summer one adds, "Morid hatal" between Pesach and Shemini Atseret; whereas in winter during Shemini Atseret and Pesach one adds, "Mashib haruach umorid hageshem" before continuing with,...Mechalkel chayim bechesed...". On what day exactly should one start with, "Moshib haruach..."? On the day of Pesach or on the day after pesach? And on what day exactly should one start with, "Morid hatal"? On the day of Shemini Atseret or the day after?
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This is dealt with in the gemara in maseches taanis. We start the mention of mashiv haruach in the chazan's repetition of shemonah esrei for Musaf of Shmini Atzeres. Once it is started, the gemoro states that even when we have a second day of Yom Tov (Simchas Torah), we continue without a break. The question had been whether to stop after mincha until musaf of simchas torah because of sefeika deyoma. However, the gemoro answers that it would be a bizayon for the chag and would indicate that the first day had not really been Yom Tov.
Similarly, we stop saying it at the chazaras hashatz of Musaf for the first day of Pesach. Shacharis and the silent Shemonah Esrai for Musaf follow the rules that had been done until that time. That is, on Pesach, say mashiv haruach vemorid hageshem during shacharis and the silent shemonah esrai of musaf. Do not say it for shacharis and the silent shemonah esrai of musaf of Shemini Atzeres.
The gemoro discusses this in terms of the shliach tzibur saying the shemoneh esrai. It does not mention the silent shemoneh esrai at all. Thus we cannot draw a conclusion as to what is done during the silent shemonah esrai from the gemoroh.
Note that this is the way it is explained in some of the Art Scroll editions of the siddur. In others (of that siddur), and in the Koren siddur as well as others,for example, the switch depends on whether or not the gabbai can announce the switch before the silent shemonah esrei or not. One can announce that mashiv haruach umorid hagashem is being said, in which case it can be said during the silent shemonah esrai. This is because the announcement of the gabbai is the equivalent informing the kahal what is to be said during all following shemoneh esrai payers. If the announcement had not been done, then the kahal would have to wait until after tefilas hageshem during the repetition of the shonah esrai.
However, one cannot announce that the kahal is to stop saying something or is not to say anything. Thus, since Ashkenazim in chutz la'aretz do not say anything during the summer, Mashiv haruach is said during the silent shemoneh esrai and dropped with tefilas tal which is said by the chazan during the repetition.
Most ashkenazim in chutz la'aretz do not say morid hatal at all and continue directly with mechalkel chaim.
I do not have experience with those that say Morid Hatal, but the logic (of the gabbai's announcement) should be the same. Thanks to @DanF and @msh210 for confirming that this logic is followed by Eidot Hamizrach and nusach 'sfarad'.