No, not without your permission. The photographer is the author of your wedding photos. They are the one who chooses the framing, takes the photo and edits it. But the faces in those photos are yours. The photographer has no rights at all over your image, only over the photos they have created for you. So, to show them on their website, on Instagram or on any other social network, they first have to ask you. And the right thing is for that permission to be clear and in writing. A line such as “unless you say otherwise, I may publish them” should not count as a yes.
Your guests have a say too
The same goes for your guests. Your grandmother, your friends and the children in the family also have rights over their own image. Even if you have signed a contract with the photographer, that does not give them permission to publish photos of your guests. If a minor appears, permission is given by their parents or guardians. If a guest sees a photo of themselves published without having authorised it, they can ask for it to be deleted or report it on the social network itself.
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What you can do with the photos
Now, the other side of the coin. Because the photographer is the author, you cannot do whatever you like with the photos either. The most common arrangement is to receive a personal-use licence. Printing them, making an album, giving copies to family or sharing them with your loved ones is usually included. Sending them to a competition or a magazine, on the other hand, usually requires the photographer's permission.
Why photographers show weddings
So why do photographers ask to show wedding photos? Because without them, the couples who come after you would have no way of seeing how they work. They would have to book blind. It is perfectly normal that not every couple wants their photos to be seen. It is your decision. A balanced option is to allow part of the coverage to be shown as a sample of their work, and nothing more. No competitions or stock photo libraries, and no other use you have not agreed to.
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What to do now
What should you do now? Get out your contract and look for the part about publishing the photos and videos. If it isn't there, or isn't clear, ask the photographer directly. Tell them what you accept and what you don't, and ask them to put it in writing in the agreement. Ideally, talk it through before signing. Keep a copy of the contract and of any messages you have exchanged. If you have already seen a photo of yourselves published without having given permission, write to them calmly and ask them to take it down. And if you are getting married in Asturias and would like to talk this over with Evan & Natalia Simon before deciding anything about photography and video, a relaxed, no-obligation call is all it takes.
Links
More useful links: boe.es, portal-cec.consumo.gob.es.