The opposition process looks different for almost every case, and it varies from jurisdiction to jurisdiction. Generally, it goes from receiving the notification to a response window, to a cooling-off period, to an adversarial phase, and finally, the decision.
In the EU, once a notice of opposition is filed within the three-month publication window, both parties enter a two-month cooling-off period for settlement negotiations (extendable to 24 months by agreement). If no settlement is reached, each party submits observations and evidence; the EUIPO issues a decision that can be appealed to the Board of Appeal.
In the US, the TTAB administers opposition proceedings. Once a notice of opposition is filed, the parties exchange evidence and briefs over a period that typically takes one to two years for a fully contested case. Settlements at any stage are common and binding.