Officially (according to the Real Academia Española), both must be used in almost all cases. The Diccionario Panhispánico de Dudas includes a section on question marks and exclamation points, which says:
Son signos dobles, pues existe un signo de apertura y otro de cierre, que deben colocarse de forma obligatoria al comienzo y al final del enunciado correspondiente; no obstante, existen casos en los que solo se usan los signos de cierre (→ 3a y d).
3a) Los signos de cierre escritos entre paréntesis se utilizan para expresar duda (los de interrogación) o sorpresa (los de exclamación), no exentas, en la mayoría de los casos, de ironía: Tendría gracia (?) que hubiera perdido las llaves; Ha terminado los estudios con treinta años y está tan orgulloso (!).
3d) Es frecuente el uso de los signos de interrogación en la indicación de fechas dudosas, especialmente en obras de carácter enciclopédico. Se recomienda colocar ambos signos, el de apertura y el de cierre: Hernández, Gregorio (¿1576?-1636), aunque también es posible escribir únicamente el de cierre: Hernández, Gregorio (1576?-1636).
In other words, they are considered paired signs, and should always be placed both at the beginning and end, except in a couple special cases where only the closing sign is used. Those special cases include:
A single question mark or exclamation point in parentheses used to express doubt, surprise, or irony.
Question marks used to indicate that a date is unsure. In this case it's still recommended to use opening and closing question marks, but a single closing mark is accepted as well.
However, as with most areas of grammar, the official rule isn't what is practically found in all Spanish-speaking areas at all times. In informal Spanish--for example, chatting online--it's common for people to drop the inverted signs or accents on vowels likely because of the point you mentioned (it can take some effort to set up a computer to type them easily). But in written, formal Spanish, the rule is generally followed.
In terms of the practical question of typing special characters on a keyboard, see this meta question which lists some options. I can't imagine there are many devices that are impossible to configure to type special characters.