If a mutual fund sell shares for a gain, do investors need to pay capital gains tax twice?

Yes, you pay capital gains tax on ETF holdings just like mutual find holdings, but you are not "double taxed".

Say you buy a fund at the beginning of the year and sell it at the end of the year. Say also that the NAV (the value of all of its holdings, less expenses) of the fund goes up by $5 per unit over the year, and realizes and distributes $1 per unit of capital gains at the end of the year. The NAV (and hence the price per unit) of the fund is then reduced by $1. So you would pay tax on the $4 of realized capital gains from selling your holdings, and the $1 that was distributed to you.

In other words, the capital gains are either distributed to you at the end of the year, or they are reflected in the price of the fund, but not both. So the capital gains tax you pay when you sell your holdings is separate from the tax you paid on the distributions (and they are not double-counted).

When it comes to funds there are two competing concepts. There is the concept of pooling your money with other investors, known as the mutual fund. In this case, realized gains are taxable (in most countries) when they occur as one can always redeeem the current value of the mutual fund.

The other way - ETFs - is best thought of as an investment company whose shares is traded on an exchange. An ETF unit is a closed system, a unit represents a certain asset allocation, typically tracking an index, at a certain time that can be traded. Additionally, some market participants ("authorized participants") can trade in a basket of shares for units of the ETF (or vice versa) which makes sure that the ETF is always closely tracking its holding. As the value of an ETF is determined by its price on the exchange, gains are only realized by selling ETF units.

With regard to dividends, they will be either distributed by the ETF or accumulated by reinvesting them into the underlying assets. An ETF will not sell assets to pay out dividends

Mutual funds are required to distribute profits back to its shareholders at least once a year (capital gains, dividends and interest).

ETFs are considered "pass-through" investment vehicles and generally, ETFs shareholders do not incur capital gains when the ETF buys and sells shares. However, ETFs can generate shareholder capital gains in certain circumstances. For example, the ETF must rebalance due to substantial changes in the underlying benchmark.