It's commonly said that everyone does Agile differently. In my experience, it's also common to do basically whatever you want and call it Agile. It can be useful to occasionally reset and examine what canonical Agile recommends. For software specifications, it's pretty simple. Do just enough, no more.
A software requirements specification (SRS) is a description of a software system to be developed, laying out functional and non-functional requirements, and may include a set of use cases that describe interactions the users will have with the software. - Wikipedia
Historically, specifications have been used to communicate to the customer (be it an internal or external customer) what will be built. Agile is built on the principle that this is actually not in the best interest of the customer. Why?
In Agile, we build in short sprints. Documentation need only be created for the next sprint, or perhaps one additional sprint. This is known as - "Just in time Documentation"
Agile artefacts such as technical spikes and development iterations mean that high-level requirements can be considered sufficient at project initiation. - Ryan Hewitt
For agile teams, specifications exist so that the team knows what they need to build. Because we are committing only to one sprint at a time, we have no need to project long-term dates that would necessitate a full specification. What documentation does the team need to get started? A highly detailed specification for just that first sprint.
An agile team is composed of a product manager, a designer and one or more developers. This team must be empowered to design and implement their vision of a solution. A specification is definitely NOT for feedback or sign-off prior to building.
Agile requirements, on the other hand, depend on a shared understanding of the customer that is shared between the product owner, designer, and the development team. That shared understanding and empathy for the target customer unlocks hidden bandwidth for product owners. They can focus on higher-level requirements and leave implementation details to the development team, who is fully equipped to do so – again, because of that shared understanding. - Dan Radigan, Senior Agile Evangelist, Atlassian
User stories are the form that specifications take. Each user story is created in advance and placed in a backlog, but only the small set of the very next stories are flesh out in detail. Then, the level of detail is very high. Designs are included at this stage, and so are detailed descriptions of fine grained behavior like validation, individual errors messages, etc.
Though the PM owns the user story, the team itself generates the detail through a processes called grooming. User empathy is critical - while PM and design naturally represent the customer in the design process, the entire team needs to understand the customer motivation and pain points.
In the end, the team may very well not build the right thing. This is where feedback comes in - at the end of a sprint, once working software is produced and shown to the customer.
Working software over comprehensive documentation - The Agile Manifesto
In Agile, feedback is given based on working software, not specifications. The team commits to delivering and demoing working software every sprint. These demos are where feedback is generated. I've often been surprised when customers don't really know what they want until they see it.