To answer this question, you need to know a little bit about the Second Triumvirate, which was formed shortly after Caesar's assassination (in March 44 BC). This triumvirate – the Second Triumvirate – was an official, legally-established institution. The law that led to its creation was the Lex Titia, which was passed in November 43 BC. From then on, the three triumvirs (Octavian, Antony and Lepidus) could refer to themselves openly and publicly (and rightly) as III VIR RPC, which is short for TRIUMVIR REI PUBLICAE CONSTITUENDAE. See, for example, this coin, although there are hundreds of examples.

Unlike this formal, legally-established Second Triumvirate, the agreement between (primarily) Caesar and Pompey and (to a lesser extent) Crassus – now known as the First Triumvirate – wasn't established in law at all. It was simply an agreement for these three men to help each other out in the pursuit of their political goals. [What these political goals were and why these three men in particular came together at this point in time seems beyond the scope of your question.]

Was it acceptable to form (temporary) political alliances to further your political aims? Of course it was. Amicitia ('friendship') was one of the fundamental forces in Roman political life. In fact, it has been said that reciprocity was the fundamental concept in all ancient ethics, i.e. you scratch my back, I'll scratch yours.

One of the best ways we see these political alliances 'in action' is though Cicero's letters. Another way is though the kind of marriages that key men arrange for their sisters and daughters. There are many, many examples of this but think about Pompey's marriage to Caesar's daughter Julia in 59 BC, for example, or Anthony's to Octavian's sister Octavia in 40 BC.

The negotiations to form the First Triumvirate (actually an informal agreement to support each other in politics) were not carried out publicly because – why would they be?