Masked Landlord has things the wrong way round.
The simple answer is that as a tenant in common you are theoretically free to sell your share to anyone you choose. However, practically speaking it is highly unlikely that you will find anyone willing to buy half a property that is jointly owned by someone else. And even if you could they would have to be a cash buyer because no mortgage company would lend for this kind of transaction.
(This is actually just as possible if you are a joint tenant. The act of selling, or attempting to sell, severs the joint tenancy and converts it to a tenancy in common).
The complicated answer deals with trusts, which always exist when property is jointly owned. At the moment you and the other person are legal owners and trustees, and you are holding the property for yourselves as beneficial joint tenants. If you sell your half you are selling your beneficial interest, but you will remain a legal owner and trustee. Any mortgage in your name will remain in place. You would still have a say in managing the property, will be responsible for mortgage payments, and as a legal owner if you wanted to sell you would be signing the forms and the money would be paid to you and the other original joint owner. But you would have to pay the appropriate share to the person who bought your beneficial share. If someone did buy, under trust law they would not necessarily gain any benefit from the property because they would not necessarily have any entitlement to live there.