What to do with dynamically translated content sharing same urls?

[D2i]

We've just added to an originally English website, Italian and German translations. User can switch between them with right hand drop down language selection menu; then the entire page will be translated (including menu, body, footer) but the urls remain the same. The Italian page have some meta data (titles and descriptions) translated as well.

Is it going to be a significantly negative effect on SEO to have the translated pages sharing the same urls?

[effectdigital]

Yes, it is the main non-recommended architectural layout by Google

Google's official guidance is here: https://support.google.com/webmasters/answer/182192?hl=en

... scroll down until you find a table. The closest architecture to your described architecture, is the parameter-based one which Google explicitly does not recommend. If Google wouldn't recommend that; without even the shallow parameter-based signifiers, there's little hope that anything would go well

I have seen a lot of sites that try to serve different language content from the same URLs and they very rarely do well or perform at all in modern times

Read Google's advice and pick from one of their pre-defined, recommended options. Whilst translation plugins can be cheap and useful, they're usually awful for SEO