What type of advertising is common during the growth stage of the product life cycle as firms compete directly and attempt to take market share from one another?

That depends for a industrial or consumer wise.

Advertisements that are suitable for growth stage are samples of the good or product given for consumers to try. (If a good or product is in the need category, it will be popular effortlessly)
At growth: sales increase thus, an option for a product to change for the better: size, price, quality, neatness. (Doing so will beat competition)
Media research: helps to showcase your product on the web and that attracts customers.
Market intelligence: helps customers to gain an understanding of a product: details, needs, terms of use.
Opinion & attitude research: comes after market intelligence to question customers' opinions and thoughts and feelings about the product.

As far as it goes; asking yourself " is product worth the time and money?" " are customer satisfied" " does my product look suitable?"

Are very important questions.