Why would Chekhov omit such an important detail in "The Lady with the Pet Dog"?It is not so we can imagine what their lives will be.
Although you do not mention the specific detail missing in the story, I'm assuming you are asking about the open-ended conclusion of the story. The story ends with Anna and Dmitri recognizing the depth of their love and realizing that their affair will continue for years to come. How they will proceed in their relationship is left unresolved, but the future will be difficult; their hardest days together lie ahead.
The conclusion is realistic, rather than romantic. It reflects Dmitri's and Anna's individual lives and the nature of the society in which they live. Each is trapped in a loveless marriage. Dmitri bears the responsibility for supporting a wife and three children, including a young daughter. Divorce is not an option for either of them. Furthermore, remaining in an adulterous affair with Dmitri is a very dangerous situation for Anna. Women in her society are punished severely for straying from their marriages, through loss of reputation, social ostracism, and financial ruin.
There is no happy ending or sad parting in the conclusion of the story; Chekhov does not detail their parting or their specific plans for the future. It ends with Dmitri and Anna's loving each other, while being suspended in a time and place that does not allow them to pursue an open life together. Dmitri and Anna have found each other at the wrong time in the wrong place, but they will not end their affair. They will continue, even though their most difficult days are to come.
Those "difficult days" are also not characterized. Part of their difficulties will surely be the need for secrecy when they meet, but they will face other difficulties, as well. They will experience the pain of living apart and the pain of living inauthentic public lives:
". . . everything that was important, interesting, essential, everything about which [Dmitri] was sincere and never deceived himself, everything that composed the kernel of his life went on in secret, while everything that was false in him, everything that composed the husk in which he hid himself and the truth which was in him--was on the surface."
Dmitri and Anna both know "that the end was still far, far away," but they also know that "the hardest, the most complicated part was only just beginning." We do not need specific details to understand what they face; their future together is clear.