You can offer such services as a private person, but you will have to disclose your address.
First, let's discuss whether this is business activity that needs to be registered. A business activity (gewerbliche Tätigkeit) is any permanent/repeated activity that is intended to make a profit. Any repeated income can be evidence of business activity, unless it is one-off or e.g. from employment that is automatically taxed anyway. This means that e.g. running ads on your website, or asking for “donations”, even if they only pay a fraction of your hosting costs, could make it necessary to register a business. So could offering a free service with the intention to attract a later customer base.
You as a natural person have the right to enter contracts, and it is not necessary to create a legal entity/corporation. However, this also means that you are the contract partner, and are personally liable. Careful: a personal Rechtschutzversicherung might not cover disputes arising from running such a service.
There are two relevant laws that require you to disclose your identity and address when running a service on the internet: the imprint requirement per §5 TMG, and the privacy policy per Art 13 GDPR.
§5 TMG establishes an imprint requirement (Impressumspflicht). You have to provide your name and the address where you live, and a quick electronic contact method such as an email address. This information must be easy to find.
This obligation applies to “geschäftsmäßige, in der Regel gegen Entgelt angebotene Telemedien” (roughly: business-like, usually paid tele-media). In practice nearly all German websites, internet services, and apps fall under this obligation. In contrast to gewerblich, geschäftsmäßig does not require a profit motive.
Art 13 GDPR lists items that you have to cover in a privacy policy. Such policies usually include the items from §15 as well. Art 13(1)(a) requires you to provide the identity and contact details of the data controller (you). This does not strictly require an address for a natural person, but an address could still be argued to be necessary in order to establish the clear identity. In any case, §5 TMG clarifies this requirement for the case of tele-media from German data controllers.
Note that when using a US-based server, you have additional GDPR compliance obligations. Regardless of where the hoster is based, you need a data processing agreement per Art 28 GDPR. For transfers of personal data outside of the EEA, you need a legal basis per Art 44, ideally an adequacy decision from the European Commission per Art 45 or standard contractual clauses per Art 46. U.S.-based companies fall under an adequacy decision iff they have self-certified under the Privacy Shield program. You also have to disclose international transfers in your privacy policy per Art 13(1)(f).
