It is obligatory to make the intention either when setting aside the amount to be given for Sadaqah al Fitr or when giving it to someone. It is through this intention that the amount is determined. Conversely, if a person gives some money to a needy person without the intention of giving Sadaqah al Fitr and later makes the intention for Sadaqah al Fitr, the obligation will not be fulfilled and Sadaqah al Fitr will have to be given again.
That being said, if a person makes the intention while the amount given as Sadaqah al Fitr is still in the possession of the receiver as is, then the obligation will be fulfilled. However, if the intention is made after the receiver has spent the money, the obligation will not be fulfilled.
If you meant something else by “specifying an amount” please explain further, we will be much obliged to provide another response based on your elaboration.