In summary, whose permission you needed depended on where you're getting your water from, and whose land your water pipes pass through.

For starters, certainly it did seem common to connect houses to water supplies, because there are many intricate rules governing situations that arise from it. For instance:

If your water pipes are laid against my house and cause me loss, an actio in factum [a legal action where damages are awarded discretionarily by the judge] is available

(Pomponius, ad Sabinum, book 10, recorded in Digest 8.2.18)

Also part of a building [which is sold] are the pipes joined to its fountains, even though they run out far beyond the structure; likewise, the water channels.

(Ulpian, ad Edictum, book 32, recorded in Digest 19.1.15)

This brings us to the question of where the water is channelled from/the source of the water.

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If you're drawing water from/through someone else's land (e.g. connecting pipes to a spring on your neighbour's land), you'd need a servitude (essentially a privilege that attaches to land) over that land known as aquaeductus (Ulpian, Institutes, Book 2, recorded in Digest 8.3.1). A servitude was generally acquired by purchasing it from your neighbour (though there were other methods; for more information on servitudes, see e.g. Barry Nicholas, An Introduction to Roman Law (Clarendon Press 1962), 141).

This was evidently a very important and common mechanism because of the many intricate rules governing it, for instance:

The right to channel water or to draw water to be conducted by a watercourse across the same ground can, in fact, be granted to several individuals. It can even be stated that the right is to be exercised on different days or at different times.

(Neratius, Rules, Book 4, recorded in Digest 8.3.2)

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If you're drawing water from a public river, you don't need any special permission unless you're channelling the water (i.e. passing pipes) over someone else's land:

As Neratius states in the third book of his Parchments, if a man is granted the right to draw water and a right of access for the purpose, he will have them both; if he is granted only the right to draw water, a right of access is also included; if he is only granted access to a spring, a right to draw water is included. These rules apply to the drawing of water from a private spring. However, with regard to a public river, Neratius tells us in the same book that, although an iter [right of way/physical access] to it must be granted, a right to draw water from it need not be...

(Ulpian, ad Edictum, Book 17, recorded in Digest 8.3.3.3)

This follows from (Marcian, Institutes, Book 3, recorded in Digest 1.8.2):

And indeed by natural law the following belong in common to all men: air, flowing water, and the sea, and therewith the shores of the sea.

However, this was subject to "reasonable use" restrictions:

The Emperors Antoninus Augustus and Verus Augustus laid down in a rescript that for the purposes of irrigating fields, water from a public river ought to be allotted in proportion to the size of those fields, unless anyone could establish that he should be allowed more than his proportionate share because of some special right of his. They further laid down that a man is only permitted to channel water if this can be done without wrong to another.

(Papirius Justus, Imperial Rulings, book 1, recorded in Digest 8.3.17)

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What about drawing water from a public aqueduct (the OP's question)? It seems that permission from the Emperor is required.

In respect of "daily water" (water that people are entitled to use everyday),

41. But in that case, it is permitted to draw the water from... the watercourse or out of any other public place. 42. This is a concession of the emperor's. No one else is entitled to grant the right to draw off water. 43. And it is sometimes granted to landed estates, sometimes to persons. Grants to landed estates are not extinguished with the person; grants to persons are lost with the person and are not transferred to another owner of the lands or to an heir or successor of any kind...

(Ulpian, ad Edictum, book 70, recorded in Digest 43.20.1.41-43)

This seems to be supported by:

Interdicts [essentially, licences] will be granted for the repair of watercourses without inquiry about whether the person taking the action is entitled to draw off the water... since if watercourses were not repaired, all use of the water would be lost and people would die of thirst. [This applies to both public and private watercourses: see D 43.21.3.5]

(Venuleius, Interdicts, book 1, recorded in D 43.21.4), which implies that you do need prior permission/entitlement to draw off water from a public watercourse.

This is consistent with the general tenor that all public infrastructure (including roadways and intercourses) was the remit of the emperor. Thus, if you wanted to channel a private waterway over public land, you'd need the emperor's permission too:

Should public ground or a public roadway lie between two estates, a servitude giving the right to draw water can still be created, although a servitude giving the right to channel water cannot. It is, however, the practice to petition the emperor for permission to lead a watercourse across a public roadway, as long as the public is not inconvenienced thereby.

(Paul, ad Sabinum, book 15, recorded in D 8.1.14.2)

From the above it seems that you needed the permission of the emperor to build an aqueduct across a public roadway, or to draw water from a public aqueduct. This is because public roadways and aqueducts belonged to the emperor, in contrast to seas and rivers which are natural, and as such "belong in common to all men" (see D 1.8.2 above)).

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In summary, whose permission you needed depended on where you're getting your water from, and whose land your water pipes pass through.

Source: The Digest of Justinian, translated by Alan Watson (University of Pennsylvania Press, 2009) (n.b. this is the main surviving primary source for Roman law, compiled on the initiative of the Emperor Justinian in the early 6th century CE)