Let me first clarify what is meant by ash-shighar marriage and Islam's stance on it. Ash-shighar marriage means that two men (for example, two brothers) marry two women (for example, two sisters) from the other family under the condition that there is no mahr (dower paid to the woman). Such ash-shighar marriage will only go through if the exchange deal goes through, and this is reason enough to invalidate the two marriages altogether.
Abu Hurairah (One of the prophet's companions) reported that Allah's Messenger has forbidden ash-shighar, which means that one man says to another: Marry me your daughter and I will marry you my daughter; or marry me your sister and I will marry you my sister." (Reported by Muslim).
In another authentic Hadith, Abdullah ibn Umar stated that Allah's Messenger forbade ash-shighar, which means that one marries his daughter to another on condition that the other man gives him his daughter in marriage, without mentioning any dower." (Related by Al-Bukhari).
These two Hadiths are perfectly clear in forbidding exchange marriages altogether. It is not surprising that this is forbidden in Islam, because it is an exchange deal that looks at women as though they were commodities that can be bought and sold.
Having stated the above, I would like to stress that if the marriage you referred to in your question contains an offer and acceptance from both parties (man and woman), and each woman is entitled to receive her mahr without the exchange deal referred to above, and no condition was attached to the first marriage that the second one will follow, then such marriage is permissible and valid according to shariah.