You get 1 Atlas mission for each master for free each day (or 2 per master if you have the "Close Allies" Atlas passive node).
Upon completing a map, you have a 35% base chance to gain a random Atlas mission. There is +1% additional chance per 15 completed Awakening objectives, which with 164 maps currently on the Atlas makes +10% maximum. The "Close Allies" Atlas passive grants an additional +10% chance. The "Ally's" Watchstone prefix mod can grant up to +4% additional chance, so with 4 such Watchstones and +25% increased effect of Watchstone mods from the "Secrets of the Stones" node, that can be up to +20%. Thus you can get a total of 75% chance for a random Atlas mission on completing a map.
Atlas passive nodes such as "Master of the Atlas" and "Animal Companion" roll separately to grant additional Atlas Missions for specific masters.
Atlas passives and Watchstone mods that grant "+X% chance for Atlas missions to be Zana's" (or other specific Masters) do not affect the number of Atlas missions you receive, they only modify the relative weighting of that Master when the game is deciding which Master to award an Atlas mission for. These stats also affect the weighting of naturally/randomly spawned Masters (ie not Atlas missions), whereas the other stats mentioned do not. So for example, normally each Master has a 1/5 chance to be chosen. If you have +100% chance for Atlas missions to be Zana's (and no other such stats), then Zana will have a 2/6 chance to be chosen and the other four Masters will each have a 1/6 chance.