What does a shareholder have a share of? In theory, a share in the assets, profits, and indeed control of a company.
However the company is a legal entity in its own right. It is down to the company itself to decide: what, if any, assets to sell; what, if any, profits to disburse to shareholders as dividends rather than re-invest as capital; in general, how to run the business.
Shareholder-owned companies are run on majority rule: so investors with generally proportional holdings in the company have little influence. In reality most decsions are taken by managers. The relationship between shareholding owners and managers is an example of the PrincipalAgentProblem. Do managers have the same incentives as shareholders?
On the other hand, managers have to keep shareholders reasonably happy by releasing dividends, or at least by maintaining the expectation of future dividends.
Equity Valuation - in theory
Company A wants to raise an amount of financial capital q. How much equity will it have to sell?
Simplifying a lot, an efficient market should price a share at the present discounted value of the estimated future earnings it will bring to the investor (shareholder). Future earnings here are the future stream of dividend payments until resale + the price the investor gets for it on resale. This is called the fair value of the share. Suppose the fair value of a 1% share in the company is p. When the company issues the equity it should, if the primary market is efficient, be able to get this fair value for its shares. To raise capital q it must then sell (q/p)% of its equity.
Whether this actually happens is another question.