Hi,
It really depends upon the type of flower. Many times the flower has died off, but the base holding the flowers is still green and developing seed. The seed can be ripe though before the flower is ready to let go of the seed.
If it is a flower that I'm unsure about, I often will check one dead flower for the seed. Unripe seed is usually green or white. Once you get to know you flowers, it becomes so much easier.
About your statement"Others say to clean the seed, dry and store it." Cleaning the seed has to do with chaff that accompanies some types of seed. Some kinds have lots of chaff like zinnias. Or sometimes like with squash seed, you need to clean away all the gunk around the seed as best you can.
The drying part is that even seed that is mature on the plant and seems dry, it may still have moisture. So seeds should be stored in paper bags to dry more. Or as in the case of some seeds like squash again, you need to let them dry on paper plates before even storing in a bag to help dissipate the moisture.
About the storing, seed should be stored in a cool dry place for longest viability.
If you have questions about specific flowers, please post. Most of the time someone knows the answer.
Remy