Personally, I'd stall a bit and see which of the two products starts to the win the politics war. In these kinds of mergers, it's typical for areas like sales or support to merge together before the software developers. That's when it usually becomes more clear which product is going to win out.
If it's clear, for example, that your product is going to win out, then working on "import from other product" is higher priority than "export to other product".
It's quite likely that one of the products takes over. Maybe one product had more traction with customers, or maybe one product is losing customers. Maybe the entire sales or development team will leave within a year, leaving one product in shambles. It's quite possible that one organization will be torn apart, who knows the terms of the acquisition.
Sometimes two products are there to stay - just ask BlueYonder, SAP, Korber or plenty of other large enterprise software vendors which end up with two products (be it merchandising systems, ERPs or WMS products they have ended up with via acquisition).
That's true, though more often the case with products that have a lot of overlap, versus really being competitors in the same market. Perhaps Trello is a good example. It overlaps with Jira, but it doesn't really compete for the exact same customers in an either/or decision.
It's hard to find a good example where a merger of two products with very similar markets where both products thrived after the merger.
If it's clear, for example, that your product is going to win out, then working on "import from other product" is higher priority than "export to other product".