It would depend on the solution concept. For this purpose, the most likely solution concept would be the Shapley Value. This determines the NET payments to the members. The share of coalition costs would then be determined by the net payments and the initial distribution of costs and may not have the property you mention. 

As Sudha Balagopalan suggests, a model based on bargaining power can determine the net payoffs of members and therefore their contributions to shared cost. Arguably the Shapley value makes implicit assumptions about bargaining power (Maskin makes this argument.) Another solution concept that could be used is Schmeidler's nucleolus. It has the advantage that it is always in the core, if the core exists. My book Value Solutions in Cooperative Games, 2013, Chapter 6, proposes an extension and synthesis of Schmeidler's solution and Nash bargaining power using bargaining power parameters. This model has not been axiomatized. But the difficulty it shares with any bargaining power approach is that a large number of bargaining power parameters would have to be calibrated. All in all, the Shapley value is the most widely recognized "value solution" (though in my opinion it has some disadvantages) and has been applied to cost sharing. My textbook, Game Theory: A Nontechnical Introduction to the Analysis of Strategy third edition 2014, has a cost-sharing example at p. 392. Both books are published by World Scientific.