Learn Italian: Secondo le indiscrezioni, il capo venderebbe l'azienda. - According to rumors, the boss would reportedly sell the company.

This is the reportative (or journalistic) conditional. By using venderebbe, the speaker distances themselves from the claim, showing that it is unconfirmed hearsay. If they used the present or future tense, it would sound like a stated fact.

The English translation includes 'reportedly', but there is no adverb for that in the Italian sentence. Why?

In Italian, the idea of 'reportedly' or 'allegedly' is built directly into this specific use of the conditional mood. Venderebbe on its own tells the listener that this is unverified information, so no extra adverb is needed.

What does the word indiscrezioni mean here? Does it mean the boss did something inappropriate?

Not necessarily! While it shares a root with the English word 'indiscretion', in Italian journalism and business, indiscrezioni commonly translates to 'leaks', 'rumors', or 'unofficial reports'.

Since rumors involve doubt and uncertainty, could we use the subjunctive mood here?

No, you cannot use the subjunctive here because there is no trigger phrase (like Si dice che... or Sembra che...) to introduce a dependent clause. Because the sentence is simply a main clause starting with the preposition Secondo..., the conditional is the correct way to express that unconfirmed nuance.