A startup company in Berlin, Germany is interested to have an interview with me for a Position which states is Fully Remote. I am an Indian residing in India.
Can I work for this company completely remotely?
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The liability is mostly for the company in these scenarios. Nearly every company advertising jobs as fully remote will either have restrictions on where their employees can be located or figure out some solution to offer you some form of local contract. It could be something like a contractor/freelance contract or a work contract with a third-party “employer of record” provider. It's up to you to decide if that's similar enough to an employment contract for your taste.
Some companies with a lot of experience with this (e.g. GitLab) also rule out employment in specific countries because there is no satisfactory legal vehicle and too much liability for the company in those countries. A startup will almost certainly rely on a third-party HR solution vendor to handle all this (and your contract is likely to be with that company or one of their subcontractors, rather than directly with the startup).
If they are not, I would consider this a red flag. I would not be terribly concerned about taxes but rather about the fact that someone might discover down the line that they cannot offer you a contract or be forced to end it when they clean up their processes.
In practice, you should know that this is possible and you don't need to rule them out before the first interview. You can always ask questions (maybe ask if they have other employees in India or outside Germany in general?) and you will still have the option to pull out when they send you an actual contract and have a chance to see how it looks like.
One downside to keep in mind is that the contract you will get is likely to be less protective than a bona fide German employment contract. It will be easier to fire you than if you were employed in Germany with a full-time permanent contract (but then again, it's not worse than typical at-will US employment or the many temp work contracts that exist in Europe).