Most people do this with Fillout’s advanced forms for Airtable.

Fillout is 100% free and it gives you the ability to update Airtable records from a form, so you would give each person their own unique Fillout link to their own record.

Fillout also offers hundreds of advanced features that Airtable’s forms don’t offer, Including the ability to create custom PDF files from a form submission, accept payments on forms, pre-fetch dynamic data from an Airtable record, display Airtable lookup fields on forms, create new linked records on a form, add a login page to your form, perform math or other live calculations on your forms, collect signatures on a form, create multi-page forms with conditional paths, connect a single form to dozens of external apps simultaneously, add CAPTCHAs to your form, and much more.

I show how to use a few of the advanced features of Fillout on these 2 Airtable podcast episodes:

• Using Fillout to update Airtable records and create an eSignature approval process with PDF file creation.
• Using Fillout to create an order entry form with line items.

Otherwise, if you want to pay for your users to have Airtable accounts, then you could create an Airtable interface which is filtered by the email address of the currently logged-in user. Then, you would share that interface with them, and they could update their record that way.

Hope this helps!

If you’d like to hire an expert Airtable consultant to help you with anything Airtable-related, please feel free to contact me through my website: Airtable consultant — ScottWorld 

Try using an Interface for that.  You'd set up a 'User' type field and make sure each record was tagged with the correct user, and you could then create a filter in the Interface that only displays the records that are tagged to the logged in user:

As each user can only see their own records due to the filter, they'll also be the only ones who can edit them.  Note that this would require all of the users to be paid though

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If they're not paid members you'll need to find another way for them to edit the data, and for this you can try using Fillout (https://www.fillout.com) which allows for 1000 free submissions a month, where each record will have their own unique link via a formula field.  You'd show this field in the Interface and so each user would be able to click on the form link and make edits.  Again, since each user only sees their own records they technically will only be able to edit their own records.  If any of the form URLs get into the wild then anyone will be able to make edits with that form URL though

You could also attempt to keep it in house by using Airtable's forms, but it's more tedious to set up (https://support.airtable.com/docs/use-case-update-records-via-a-form) so if you're expecting fewer than 1k submissions may as well just use Fillout really

I’ll just add that another option you could look into is Plumsail Forms. You can create customized forms that connect directly to Airtable, allowing users to edit their records without needing paid Airtable accounts.

You can include the form link in a formula field or set up automation to send it to users by email, which helps you avoid giving them direct access to Airtable.