In the beginning of Shemot, we meet two מְיַלְּדֹת הָעִבְרִיֹּת, "Jewish midwives," or perhaps "midwives for the Jewesses." We are told their names are Shifra and Puah. To cite Shemot 1:15:
וַיֹּאמֶר מֶלֶךְ מִצְרַיִם, לַמְיַלְּדֹת הָעִבְרִיֹּת, אֲשֶׁר שֵׁם הָאַחַת שִׁפְרָה, וְשֵׁם הַשֵּׁנִית פּוּעָה.
"And the king of Egypt spoke to the Hebrew midwives, of whom the name of the one was Shiphrah, and the name of the other Puah;"
Now the name Shifra has been found in a list of Egyptian names (see W. F. Albright, “Northwest-Semitic Names in aList of Egyptian Slaves from the Eighteenth Century BC,” JAOS 74(1954): 229), which would imply that the midwives were Egyptians, who served (exclusively or among others) Jewish women. Alternatively, some Jews assumed Egyptian names (in addition to or instead of their Hebrew names), midrashim to the contrary. Alternatively, Hebrew names are on the Egyptian slave names - which would be understandable given that the Hebrews were *slaves*. Assuming they were Egyptian midwives for Jewesses, I would say they served non-Jews as well, because otherwise how would Pharoah believe them when they say that Jewish women give birth before the midwives get there - how would they make a living?
Now the midrash associates Shifra and Puah with Yocheved and Miriam (see Rashi). Why? One reason may be that Shemot 2 speaks of Moshe's mother and sister without giving their names, so Shifra and Puah and sucked into the vaccuum, to connote that they were the same folks. Alternatively, this is an example of the closed canon approach. That is, when we encounter a previously unencountered person who does not get much exposure, rather than saying that this is a person we do not know, we assume that it is someone from within the canon. This we say that Boaz, who is not mentioned anywhere in Shoftim, is identical with Ivtzan, who is mentioned in Shoftim. (see my discussion of closed canon here.
However, I had another thought. When Pharoah's daughter finds Moshe, she says, (Shemot 2:6) מִיַּלְדֵי הָעִבְרִים זֶה, "This is one of the Hebrews' children." However, the form of מִיַּלְדֵי הָעִבְרִים is similar to מְיַלְּדֹת הָעִבְרִיֹּת. Ignoring nikud, the consonants almost match. Further, if we could just double the mem, we would get מִמְּיַּלְדֵי הָעִבְרִים זֶה. This would be a pronouncement that Moshe is from the midwives of the Hebrews. This would be Yocheved, and once we know the identity of one of the midwives, it is not much of a stretch to identify Miriam as the other.
Can we just double the mem? It is not without precedent. The mishna in Horayot 3b relates the law that there is only a chiyuv korban when bet din is mekayem miktzat and oker miktzat, but if they say "Ain Avodah Zarah BaTorah," then there is no chiyuv korban. The gemara discusses sources for this din. On 4b, Ulah cites the pasuk in Vayikra 4:13:
וְאִם כָּל-עֲדַת יִשְׂרָאֵל, יִשְׁגּוּ, וְנֶעְלַם דָּבָר, מֵעֵינֵי הַקָּהָל; וְעָשׂוּ אַחַת מִכָּל-מִצְוֹת יְהוָה, אֲשֶׁר לֹא-תֵעָשֶׂינָה--וְאָשֵׁמוּ.
"And if the whole congregation of Israel shall err, the thing being hid from the eyes of the assembly, and do any of the things which the LORD hath commanded not to be done, and are guilty"
The pasuk says וְנֶעְלַם דָּבָר. Ulah says you should read in it "VeNeelam *Mi*Davar." Rashi explains that the final mem of וְנֶעְלַם we throw on דָּבָר. It is nidreshet both for the word before it and after it, so implies מִדָּבָר, of the matter, and not kol davar, the entire matter.
So a mem can be doubled and darshened. Perhaps something similar is at work here.