Sitting magistrates organised the elections. This wasn't 1-to-1. Most elections were held in various comitia – in reverse seniority (consuls first, quaestors last) – presided by a consul. Plebeian tribunes, however, were created in elections presided by other tribunes (eg Plutarch Life of Tiberius Gracchus' narrative, 13.2 and 18.1).
How the president was selected isn't entirely clear. In the middle republic the consuls were usually at war: one would dash home, do the elections, and immediately bounce; otherwise one would appoint a dictator to conduct the elections and the dictator would immediately resign. In the late republic when both consuls largely remained in the city during their terms it was more common instead to set elections in Quintilis (July) (when they were not delayed for various reasons); this meant the consul prior held elections since he held fasces in odd-numbered months. PPDR QRR (2019) pp 64ff. For tribunes, the lot is likely when they could not agree among themselves.
If there were no successors for the consuls by the end of their terms, an interrex was elected in the senate from patrician members. Each interrex held office for five days. The first interrex always resigned without doing anything; the next interrex could then hold consular elections. When the consuls were then elected, the interrex resigned and the consuls then assumed presidency of the electoral comitia. This restored the curule succession and the republic then continued.
A single exception exists for 43 BC, when Octavian and Quintus Pedius were elected by a proconsul appointed for that purpose because both consuls were dead but the auspices (the right to consult the gods) did not revert to the patres (the senate) because a patrician praetor was still in office (ironically serving with Antony during the war between the senate and Antony which had killed both consuls).
See generally:
- Lintott Constitution of the Roman republic (1999)
- Wilson Dictator (2021), re dictators comitiorum habendorum causa
- Pina Polo and Diaz Rodriguez Quaestorship in the Roman republic [PPDR QRR] (2019), re elections for the most junior magistrates
- Pelling "Triumviral period" in CAH vol 9 (2nd edn 1996), re 43 BC events, but also see MRR vol 2 for years after 44 BC
- Ramsay "How and why was Pompey made sole consul" Historia 65 (2016) pp 298–324, re the intricacies of the interreges
