I am interested to get married to my female cousin and she too is interested whilst on the other hand my male cousin is interested to get married to my sister and likewise the same for her. This is based upon the fact that the guardians of each family are pleased with it, (I mean both parents). The point is that all members from both sides are completely pleased with it and there are no conditions between us. Also, we acknowledge that the marriage of “Shighâr” is haraam. However, what we request from you O’ Noble Sheikh, (may Allaah preserve you), is that you direct us to a safe and upright way (in performing the marriage), in order that the marriage be valid, so that we will be safe from falling into the marriage of “Shighâr” and from falling into sin and unlawful acts. Also, is it necessary that each one of the spouses gives more (dowry) than the other?
In The Name of Allaah, The Most Beneficent, The Most Merciful, All praise is due to Allaah and may the peace and blessings of Allaah be upon the Messenger of Allaah and upon his family, his companions and upon those who follow his guidance.
To proceed, The marriage of “Shighâr” is as he mentioned that it is NOT permissible whereas the Prophet ﷺ, prohibited it. And it (i.e. the marriage of “Shighâr“) is to say to another “marry me off your daughter and I will marry you off my daughter” or “marry me off your sister and I will marry you off my sister” or of that sort. This is considered the marriage of “Shighâr” as it is upon preset conditions from both sides. 
If it occurs without preset conditions from both sides (while they are all pleased with it) and there are NO PRESET CONDITIONS, whereas this one is pleased and the other is pleased and the woman is pleased then it is okay and this is not considered “Shighâr“. It is so, because “Shighâr” is to have preset conditions between them that is: ‘I do not marry you off up until you marry me off’ and as for if each one of them is interested in the marriage of the other even if the other one doesn’t get married whereas he marries his sister to him and the other didn’t marry the other to his sister. Likewise, pertaining to the daughters, being that they made an agreement without any preset conditions whereas he marries his sister to him and the other marries him to his sister or his daughter while both sides are pleased with it and this is not considered “Shighâr” and it doesn’t matter if they aren’t equal (in the dowry given). So, whether the dowries are the same or not, it doesn’t harm (i.e. the contract of the marriage).”
Translated by AbuFajr AbdulFattaah bin Uthman & Edited by Umm Hasna
 As has been reported in the following Hadeeth:
حَدَّثَنَا عَبْدُ اللَّهِ بْنُ يُوسُفَ، أَخْبَرَنَا مَالِكٌ، عَنْ نَافِعٍ، عَنِ ابْنِ عُمَرَ ـ رضى الله عنهما ـ أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم نَهَى عَنِ الشِّغَارِ، وَالشِّغَارُ أَنْ يُزَوِّجَ الرَّجُلُ ابْنَتَهُ عَلَى أَنْ يُزَوِّجَهُ الآخَرُ ابْنَتَهُ، لَيْسَ بَيْنَهُمَا صَدَاقٌ.
Narrated ibn `Umar: Allah’s Messenger ﷺ forbade ash-Shighar, which means that somebody marries his daughter to somebody else, and the latter marries his daughter to the former without paying Mahr.
Agreed Upon [Source]